UNIVERSITY OF CALIFORNIA, SAN DIEGO

Logarithmic Sobolev Inequalities for the Free Loop Group A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Trevor R. Carson

Committee in charge: Professor Bruce K. Driver, Chair Professor Pat J. Fitzsimmons Professor George M. Fuller Professor Aneesh V. Manohar Professor Nolan R. Wallach

Copyright Trevor R. Carson, 1997

All rights reserved.

The dissertation of Trevor R. Carson is approved, and it is acceptable in quality and form for publication on microfilm:

Chair

University of California, San Diego

1997

TABLE OF CONTENTS

Signature Page . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

Vita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2. Definitions and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 The Geometry of the Free Loop Algebra . . . . . . . . . . . . . . . . . . . . . 7 1. The Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2. Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3. Ricci Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3 Finite Dimensional Approximations . . . . . . . . . . . . . . . . . . . . . . . . 33 1. Convergence of Finite Dimensional Laplacian . . . . . . . . . . . . . . . . . 44 2. Convergence of Finite Dimensional Ricci Tensor . . . . . . . . . . . . . . . 61

4 The Heat Kernel Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Integration By Parts on L(G) . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 1. Closed and Closable Quadratic Forms . . . . . . . . . . . . . . . . . . . . . 82 2. The Orthonormal Frame Bundle . . . . . . . . . . . . . . . . . . . . . . . . 88 3. Parallel Translation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4. Brownian Motion Via The Development Map . . . . . . . . . . . . . . . . 91 5. Integration By Parts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6. Closability of the Quadratic Form . . . . . . . . . . . . . . . . . . . . . . . 98

6 The Logarithmic Sobolev Inequality . . . . . . . . . . . . . . . . . . . . . . . . 101

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

VITA

April 30, 1966 Born, Los Angeles, California 1990 B.A., California State University, Fresno 1989,1991 Meteorological Summer Intern, National Severe

Storms Laboratory, Norman, Oklahoma

1990-1991 Graduate Assistant, California State University,

Fresno

1991-1992 Math Instructor, California State University,

Fresno

1992 M.A., California State University, Fresno 1992-1995 Summer Bridge Math Instructor at

California State University, Fresno, California State University, San Diego, and University of California, San Diego

1996 Summer Session in Probability, Institute for Advanced

Study, Princeton, New Jersey

1992-1997 Teaching Assistant, University of California, San Diego 1997 Ph.D., University of California, San Diego

ABSTRACT OF THE DISSERTATION

Logarithmic Sobolev Inequalities for the Free Loop Group

by

Trevor R. Carson Doctor of Philosophy in Mathematics University of California, San Diego, 1997 Professor Bruce K. Driver, Chair

For a compact Lie group, G, let L(G) be the free loop group consisting of the space of continuous paths g : [0, 1] → G with g(0) = g(1). For a given Ad_{G}-invariant inner product on the Lie algebra of G and a suitable related inner product on the “Lie algebra” of L(G) we derive the corresponding Riemannian structure on L(G). For this structure we have constructed finite dimensional approximations which are used in proving a logarithmic Sobolev inequality and integration by parts formulas on L(G). The underlying probability measure on L(G) is a heat kernel probability measure which can be derived by an L(G)-valued Brownian motion.

Chapter 1

Introduction

1.1 Overview

The purpose of this paper is two-fold. As the title indicates, one major purpose

is to prove a certain logarithmic Sobolev inequality which is of the form

∫_{L} f^{2} log f^{2}dν_{t} ≤ \frac{2}{C} (e^{Ct} − 1)E(f) + ∫_{L} f^{2}dν_{t} log(∫_{L} f^{2}dν_{t}), t > 0,  (1.1)

where L is a particular infinite dimensional Lie group called the free loop group, ν_{t} is a probability measure on L depending on t and a fixed element of L, C is a constant, and E is a particular quadratic form. The other purpose of the work that follows is to develop the structure of the free loop group itself.

We start by developing a Riemannian geometric structure on L, which includes

a Levi-Civita covariant derivative and associated curvature and Ricci tensors. We then devote a considerable amount of effort to showing how elements of the geometry of L, in particular, the Laplacian and Ricci tensor, can be approximated by corresponding elements of finite dimensional subspaces. These approximations will be important tools in the proof of the logarithmic Sobolev inequality.

Next, we construct a particular probability measure, ν_{t}, which is the “heat

kernel” measure on L used for integration in (1.1). The construction of ν_{t} requires methods from stochastic analysis on manifolds. In particular, we show how to derive a Brownian motion stochastic process, Σ, on L by using the notion of parallel translation and the “rolling map” from a flat vector space to the manifold, L. The measure ν_{t} is simply defined to be the distribution of Σ at time t.

We then show that the quadratic form, E, appearing in (1.1) is closable. By

extending E to its closure, we are able to prove that the inequality holds on a class of functions larger than that of the original domain of E. The key to proving E is closable is the use of an integration by parts formula on L, which in turn relies on the aforementioned finite dimensional geometrical approximations.

Finally, in proving the inequality itself, we choose a class of “cylinder” functions

which depend only on finite dimensional subspaces of L (these comprise the domain of E). For such a function, f, the inequality can be realized on a finite dimensional manifold for which the logarithmic Sobolev inequality is already know to hold when the corresponding Ricci tensor is bounded below (this bound is the constant, C , appearing in (1.1)). In this sense, it is appropriate to say that we are really extending a know inequality to an infinite dimensional space.

The idea of such an extension to an infinite dimensional space was first pioneered

by B. K. Driver and T. Lohrenz [5] in their work on the pinned loop group. I owe much gratitude to Dr. Driver for his guidance and suggestions during the work of this paper, and for his altruistic sharing of time, patience, and knowledge.

1.2 Definitions and Results

In this section we give a more detailed summary of the mathematical structures

and mappings that serve as a context for this paper, as well as the key results that will arise.

For the entirety of the paper, we assume that G is a compact, connected Lie

group. In particular, G is a C^{∞} manifold endowed with a group structure such that the group operations of multiplication and inverses are smooth. Furthermore, we assume that G has a Riemannian structure consisting of an inner product, ⟨·, ·⟩, on the Lie algebra, 𝔤, of G (by definition, 𝔤 ≡ T_{e}G is the tangent space of G at the identity e). This inner product is assumed to be Ad_{G}-invariant, which implies that the adjoint operator, ad_{A}, is skew-symmetric for each A ∈ 𝔤. Note that Ad : G → Aut(𝔤) is the adjoint representation on G where, for each σ ∈ G, Ad_{σ} is the restriction to 𝔤 of the differential of the inner automorphism τ ↦ στσ^{−1}. The adjoint operator ad : 𝔤 → End(𝔤) is the restriction to 𝔤 of the differential of Ad. It turns out that for each A, B ∈ 𝔤, ad_{A}B = [A, B] where [A, B] ≡ AB − BA is the Lie bracket on 𝔤.

As an example, one may take G ≡ SO(n), the set of n × n real orthogonal

matrices with determinant 1. In this case, the Lie algebra 𝔤 = T_{e}G turns out to be 𝔤 = so(n), the set of n × n real matrices which are skew symmetric. A natural choice for an inner product would be ⟨A, B⟩ ≡ tr(A^{T} B) for all A, B ∈ so(n).

The free loop group on G is defined to be the space of continuous loops:

L(G) ≡ {g : [0, 1] → G : g is continuous and g(0) = g(1)}.

Note that L(G) is itself an infinite dimensional topological group under the pointwise multiplication operation:

(g_{1}g_{2})(s) ≡ g_{1}(s)g_{2}(s) ∀ g_{1}, g_{2} ∈ L(G), s ∈ [0, 1].

We will take the topology on L(G) to be that of uniform convergence.

Embedded in L(G) is the space of finite energy free loops, H(G). We will define

the Lie algebra of H(G) to be the Cameron-Martin Hilbert space, H(𝔤), consisting of all absolutely continuous functions h : [0, 1] → 𝔤 such that h(0) = h(1) and (h, h) < ∞, where (·, ·) is the inner product on H defined by

(h, k) ≡ ∫_{0}^{1}{⟨h′(s), k′(s)⟩ + ⟨h(s), k(s)⟩}ds

for h, k ∈ H.

The first step we take in the development of the geometry of L(G) is to find

the Levi-Civita covariant derivative on H(𝔤). By definition, this is the unique covariant derivative ∇ : H(𝔤) ⊗ H(𝔤) → H(𝔤) that is metric compatible and torsion free, i.e.,

(∇_{h}k, l) + (k, ∇_{h}l) = 0
and ∇_{h}k − ∇_{k}h = [h, k]

for all h, k, l ≡ H(𝔤). The explicit formula for ∇ turns out be

∇_{h}l(s) = − ∫_{0}^{1} G_{s}(s, t)[h, l′](t)dt + \frac{1}{2} ∫_{0}^{1} G(s, t)[h, l](t)dt,

where G(s, t) is the Green’s function associated with the differential equation

−u′′ + u = f

with certain periodic boundary conditions (see the proof of Lemma 2.1.7). It is important to note that this Green’s function is what characterizes the majority of the development of the geometry of the free loop group and epitomizes the distinction between the free loop group and the pinned loop group. Its properties play a crucial role in many of our calculations.

Based on ∇, we derive explicit formulas for the curvature tensor R : H(𝔤) ⊗

H(𝔤) ⊗ H(𝔤) → H(𝔤), defined by

R⟨h, k⟩l ≡ ∇_{h}∇_{k}l − ∇_{k}∇_{h}l − ∇_{[h,k]}l,

and the Ricci tensor Ric : H(𝔤) ⊗ H(𝔤) → R, defined by

Ric⟨h, l⟩ ≡ ∑_{k∈S}(R⟨h, k⟩k, l),

where S is an orthonormal basis of H(𝔤). The curvature tensor is given in Theorem 2.2.1 and the Ricci tensor is given in (2.21). An important property of the Ricci tensor we derive is that it is bounded below (see Theorem 2.3.11 ) in the sense that there exists a constant C > 0 such that

Ric⟨h, h⟩ ≥ −C⟨h, h⟩ ∀ h ∈ H(𝔤).

We next show the existence of a heat kernel probability measure on L(G). By

definition, this is a family of measures {ν_{t}(g_{0}, ·)}_{t>0,g_{0}∈L(G)} such that

u(t, g_{0}) ≡ ∫_{L(G)} f(g)ν_{t}(g_{0}, dg)

solves the heat equation

\frac{∂u}{∂t} = \frac{1}{2}Δu, lim_{t↓0} u(t, g_{0}) = f(g_{0})

for all f ∈ FC^{∞}(L(G)), where FC^{∞}(L(G)) are the cylinder functions on L(G) of the form f = F ∘ π_{P} such that P is a finite partition of [0, 1] and π_{P} : L(G) → G^{P} is the projection restricting to partition points.

To obtain the heat kernel measure, we first construct an L(G) -valued Brownian

motion, {Σ_{t}}_{t≥0}. In Euclidean space, a Brownian motion is usually defined to be a continuous random process with independent, mean-zero Gaussian increments. In a more abstract space, such as L(G), we use the martingale characterization of Brownian motion found in Theorem 4.0.2 which requires {Σ_{t}}_{t≥0} to be a diffusion process with generator \frac{1}{2}Δ. We start, however, with a different Brownian motion, \{β_{t}\}_{t≥0}, which is valued in a vector space (an abstract Wiener space) containing our Cameron-Martin Hilbert space H(𝔤), and is well-known to exist. {Σ_{t}}_{t≥0} is then the image of {β_{t}}_{t≥0} under the rolling map characterized by the property that the “velocity vector” of Σ at time t is the image of the “velocity vector” of β at time t under parallel translation along Σ. Since Brownian motions are almost nowhere differentiable however, we do not have the luxury of velocity vectors. We rely instead on the use of stochastic differential equations as given in Theorem 4.0.1. By taking ν_{t}(g_{0}, ·) to be the law of g_{0}Σ at time t, we obtain our desired heat kernel measure (see Theorem 4.0.3).

For each t > 0 and g_{0} ∈ L(G), the logarithmic Sobolev inequality in (1.1) can

then be verified for all cylinder functions f ∈ FC^{∞}(L(G)) using the measure ν_{t}(g_{0}, ·). The quadratic form E depends on t and g_{0} and is defined by

E(f) = E_{g_{0},t}(f) ≡ ∫_{L(G)} ∥∇f(g)∥_{H(𝔤)}^{2} ν_{t}(g_{0}, dg).  (1.2)

The key to proving the inequality is to note that for a given f = F ∘ π_{P} and partition P of [0, 1], the measure ν_{t}(g_{0}, ·) is restricted to a corresponding measure on the finite dimensional compact manifold G^{P} on which it becomes absolutely continuous with respect to Riemannian volume measure and has an associated heat kernel density function. In this finite dimensional setting of G^{P}, the inequality is known to hold provided the corresponding Ricci tensor, Ric_{P}, is bounded below by some constant −C_{P}. By proving the finite dimensional approximations to the Ricci tensor on H(𝔤) found in Theorem 3.2.1, we are able to show that the finite dimensional Ricci tensors, Ric_{P}, can be bounded below by a constant, −C, which is independent of the partitions P.

Finally, we prove that the quadratic forms E_{g_{0},t} are closable, which means

that each E_{g_{0},t} can be extended to a closed form, \overline{E}_{g_{0},t}, on a Hilbert space containing FC^{∞}(L(G)). Thus, the inequality can be extended to this larger class of functions making it a truly infinite dimensional theorem. As we have mentioned before, the closability of E_{g_{0},t} depends on an integration by parts formula (Theorem 5.20). Because of the dependence of E_{g_{0},t} on the gradient ∇ in (1.2), E_{g_{0},t} is closable if and only if ∇ is a closable operator. This holds precisely when ∇ has a densely defined adjoint, which we demonstrate to be the case in Theorem 5.6.1.

The Geometry of the Free Loop

Algebra

2.1 The Covariant Derivative

In this section we wish to derive the “Levi-Civita covariant derivative” on H(𝔤).

Since H(𝔤) is equivalent to the left-invariant vector fields on L(G) (as illustrated by the next definition), we start by deriving the Levi-Civita covariant derivative in the context of the left-invariant vector fields.

Definition 2.1.1 For h ∈ T_{e}L, define h̃ to be the left-invariant vector field on L corresponding to h:

h̃(g) ≡ L_{g∗}h ∀g ∈ L,
i.e., (h̃(g))(s) = L_{g(s)∗}h(s) ∀s ∈ [0, 1].

Definition 2.1.2 Define an inner product, also called (·, ·), on left-invariant vector fields by

(h̃(g), k̃(g)) = (L_{g∗}h, L_{g∗}k)
≡ (h, k) ∀g ∈ L

where the last inner product is the one on H. Note that (·, ·) is left- invariant.

Now let ∇ denote the Levi-Civita covariant derivative, with respect to (·, ·),

which is uniquely determined on left-invariant vector fields. To find an explicit formula for ∇, we thus need to compute ∇h̃k̃ for all h, k ∈ H.

Using the requirements that ∇ be metric compatible and torsion free, by (a)

on p.55 of Do Carmo [7], we must have, for all h, k, l ∈ H,

(k̃, ∇h̃l̃) = \frac{1}{2}\{l̃(h̃, k̃) + h̃(k̃, l̃) − k̃(l̃, h̃) − (k̃, [l̃, h̃]) − (l̃, [h̃, k̃]) − (h̃, [l̃, k̃])\}
= −\frac{1}{2}\{(k̃, [l̃, h̃]) + (l̃, [h̃, k̃]) + (h̃, [l̃, k̃])\}
= −\frac{1}{2}\{(k̃, [l, h]^{}˜ ) + (l̃, (ad_{h}k)^{}˜ ) + (h̃, (ad_{l}k)^{}˜ )\}
= −\frac{1}{2}\{(k, [l, h]) + (l, ad_{h}k) + (h, ad_{l}k)\}
= −\frac{1}{2}\{(k, [l, h]) + (ad_{h}^{∗}l, k) + (ad_{l}^{∗}h, k)
= −\frac{1}{2}\{(k̃, [l, h]^{}˜ ) + ((ad_{h}^{∗}l)^{}˜ , k̃) + ((ad_{l}^{∗}h)^{}˜ , k̃),

where ad_{h}^{∗} is the adjoint of ad_{h} : H(𝔤) → H(𝔤)^{∗} with respect to (·, ·). In the above calculations, we are using the well-know fact from Lie group theory that if h(s) and k(s) are in 𝔤, then [h̃(s), k̃(s)] = [h(s), k(s)]^{}˜ . Note that l̃(h̃, k̃) = 0 since (h̃, k̃) is constant over L. Since k̃ runs over all left-invariant vector fields, then

∇h̃l̃≡ −\frac{1}{2}\{[l, h] + ad_{h}^{∗}l + ad_{l}^{∗}h\}^{}˜ .  (2.1)

The existence of ad_{h}^{∗} for each h ∈ H(𝔤) follows by Riesz’s representation theo-

rem on Hilbert spaces. This theorem depends on ad_{h} being a bounded operator which is the case as follows. For all h, k ∈ H(𝔤),

∥ad_{h}k∥_{H(𝔤)}^{2} = ∫_{0}^{1}(∥[h, k]′(s)∥_{𝔤}^{2} + ∥[h, k](s)∥_{𝔤}^{2})ds
= ∫_{0}^{1}(∥[h′, k](s) + [h, k′](s)∥_{𝔤}^{2} + ∥[h, k](s)∥_{𝔤}^{2})ds
≤ ∫_{0}^{1} c(2 ∥h′(s)∥_{𝔤}^{2} ∥k(s)∥_{𝔤}^{2} + 2 ∥h(s)∥_{𝔤}^{2} ∥k′(s)∥_{𝔤}^{2} + ∥h(s)∥_{𝔤}^{2} ∥k(s)∥_{𝔤}^{2})ds
≤ c(2M_{1} + 3M_{2}) ∥k∥_{H(𝔤)}^{2} ,

where c is a constant depending on [·, ·], M_{1} ≡ sup_{s∈[0,1]} ∥h′(s)∥_{𝔤}^{2}, M_{2} ≡ sup_{s∈[0,1]} ∥h(s)∥_{𝔤}^{2}.

Let us now determine ad_{h}^{∗} for h ∈ H(𝔤). For convenience in calculations, we

will assume for the time being that [h, l′] and [l, h′] are in H(𝔤) so that the derivatives [h, l′]′ and [l, h′]′ exist almost everywhere and [h, l′](0) = [h, l′](1) and [l, h′](0) = [l, h′](1). However, we will show that our final expression for ∇_{h}l will not require [h, l′] and [l, h′] to be differentiable almost everywhere or periodic.

Proposition 2.1.3 Let h, l ∈ H(𝔤) such that [h, l′] ∈ H(𝔤). Then

(ad_{h}^{∗}l)(s) = ∫_{0}^{1} a cosh(r(s, t))([l, h] + [l′, h′] + [h, l′]′)(t)dt

where a = _{\frac{1}{2 sinh( \frac{1}{2} )}} and

r(s, t) = \{ \begin{matrix}s − t − \frac{1}{2} & 0 ≤ t < s \\ s − t + \frac{1}{2} & s ≤ t ≤ 1. \end{matrix}

Proof. First note that by Riesz’s Theorem, since (l, [h, ·]) is a bounded linear operator (being continuous), ad_{h}^{∗}l exists and is unique for all l ∈ H(𝔤).

Let k ∈ H(𝔤). Then ad_{h}^{∗} satisfies

(ad_{h}^{∗}l, k) = (l, [h, k])
= ∫_{0}^{1}{⟨l′, [h, k]′⟩ + ⟨l, [h, k]⟩}ds
= ∫_{0}^{1}{⟨l′, [h′, k]⟩ + ⟨l′, [h, k′]⟩ + ⟨l, [h, k]⟩}ds
= ∫_{0}^{1}{−⟨[h′, l′], k⟩ − ⟨[h, l′], k′⟩ − ⟨[h, l], k⟩}ds  (2.2)

Since (ad_{h}^{∗}l, k) = ∫_{0}^{1}{⟨(ad_{h}^{∗}l)′, k′⟩ + ⟨ad_{h}^{∗}l, k⟩}ds, we would like to write (2.2) in the form of

(u, k) = ∫_{0}^{1}{⟨u′, k′⟩ + ⟨u, k⟩}ds

for some u ∈ H(𝔤). By the uniqueness of ad_{h}^{∗}l, then we would have ad_{h}^{∗}l = u.

Let us first note that using integration by parts on the second term in (2.2)

gives

∫_{0}^{1} −⟨[h, l′], k′⟩ds = −⟨[h, l′], k⟩|_{0}^{1} − ∫_{0}^{1} −⟨[h, l′]′, k⟩ds
= ∫_{0}^{1}⟨[h, l′]′, k⟩ds.

So we may rewrite (2.2) as

∫_{0}^{1}⟨[l, h] + [l′, h′] + [h, l′]′, k⟩ds.  (2.3)

At this point we need to prove some facts which we will state as lemmas since

they will be useful to us for future calculations. For what follows, let

G(s, t) ≡ a cosh(r(s, t))  (2.4)
and F (s, t) ≡ G_{s}(s, t) = −G_{t}(s, t)
= a sinh(r(s, t)) ∀ (s, t) ∈ [0, 1]^{2},

where a and r(s, t) are defined as in Proposition 2.1.3.

Lemma 2.1.4 For f ∈ L^{1}([0, 1], 𝔤) or f ∈ L^{1}([0, 1], R), u(s) ≡ ∫_{0}^{1} G(s, t)f(t)dt is absolutely continuous and has the following derivative:

u′(s) = ∫_{0}^{1} G(s, t)f(t)dt

for almost all s ∈ [0, 1].

Proof: Note that

∫_{0}^{1} G(s, t)f(t)dt  (2.5)
= ∫_{0}^{s} a cosh(s − t − \frac{1}{2})f(t)dt + ∫_{s}^{1} a cosh(s − t + \frac{1}{2})f(t)dt
= a cosh(s) ∫_{0}^{s} cosh(−t − \frac{1}{2})f(t)dt + a sinh(s) ∫_{0}^{s} sinh(−t − \frac{1}{2})f(t)dt
+a cosh(s) ∫_{s}^{1} cosh(−t + \frac{1}{2})f(t)dt + a sinh(s) ∫_{s}^{1} sinh(−t + \frac{1}{2})f(t)dt. (2.6)

Since cosh is bounded on [0, 1] and f ∈ L^{1}([0, 1]), then ∫_{0}^{s} cosh(−t − \frac{1}{2})f(t)dt

is an indefinite integral of an L^{1} function and is thus absolutely continuous by the fundamental theorem of calculus for Lebesgue integrals (see Folland [8], P.102, or Royden [19], P.110). Since cosh is absolutely continuous, then a cosh(s) ∫_{0}^{s} cosh(−t − \frac{1}{2})f(t)dt is also absolutely continuous being the product of two absolutely continuous functions (see Royden [19], p. 111).

To take the derivative in (2.6) we use the product rule and the fundamental

theorem of calculus as follows:

u′(s)
= a sinh(s) ∫_{0}^{s} cosh(−t − \frac{1}{2})f(t)dt + a cosh(s) cosh(−s − \frac{1}{2})f(s)
+a cosh(s) ∫_{0}^{s} sinh(−t − \frac{1}{2})f(t)dt + a sinh(s) sinh(−s − \frac{1}{2})f(s)

+a sinh(s) ∫_{s}^{1} cosh(−t + \frac{1}{2})f(t)dt − a cosh(s) cosh(−s + \frac{1}{2})f(s)
+a cosh(s) ∫_{s}^{1} sinh(−t + \frac{1}{2})f(t)dt − a sinh(s) sinh(−s + \frac{1}{2})f(s)
= ∫_{0}^{s} a sinh(s − t − \frac{1}{2})f(t)dt + ∫_{s}^{1} a sinh(s − t + \frac{1}{2})f(t)dt
+a cosh(−\frac{1}{2}) − a cosh(\frac{1}{2})
= ∫_{0}^{1} G(s, t)f(t)dt

for almost all s ∈ [0, 1]. □

Lemma 2.1.5 For f ∈ L^{1}([0, 1], 𝔤) or f ∈ L^{1}([0, 1], R),

\frac{d}{ds}(∫_{0}^{1} F (s, t)f(t)dt) = −f(s) + ∫_{0}^{1} G(s, t)f(t)dt

for almost all s [0, 1]. (If u(s) ≡ ∫_{0}^{1} G(s, t)f(t)dt, then u′′(s) = −f(s) + ∫_{0}^{1} G(s, t)f(t)dt for almost all s ∈ [0, 1].)

Proof: We first note that

∫_{0}^{1} F (s, t)f(t)dt
= a sinh(s) ∫_{0}^{s} cosh(−t − \frac{1}{2})f(t)dt + a cosh(s) ∫_{0}^{s} sinh(−t − \frac{1}{2})f(t)dt
+a sinh(s) ∫_{s}^{1} cosh(−t + \frac{1}{2})f(t)dt + a cosh(s) ∫_{s}^{1} sinh(−t + \frac{1}{2})f(t)dt.

By the same reasoning used in Lemma 2.1.4, it is clear that ∫_{0}^{1} F (s, t)f(t)dt is absolutely continuous. Recalling that a = _{\frac{1}{2 sinh( \frac{1}{2} )}}, we then have

\frac{d}{ds}(∫_{0}^{1} F (s, t)f(t)dt)

= a cosh(s) ∫_{0}^{s} cosh(−t − \frac{1}{2})f(t)dt + a sinh(s) cosh(−s − \frac{1}{2})f(s)

+a sinh(s) ∫_{0}^{s} sinh(−t − \frac{1}{2})f(t)dt + a cosh(s) sinh(−s − \frac{1}{2})f(s)

+a cosh(s) ∫_{s}^{1} cosh(−t + \frac{1}{2})f(t)dt − a sinh(s) cosh(−s + \frac{1}{2})f(s)
+a sinh(s) ∫_{s}^{1} sinh(−t + \frac{1}{2})f(t)dt − a cosh(s) sinh(−s + \frac{1}{2})f(s)
= ∫_{0}^{s} a cosh(s − t − \frac{1}{2})f(t)dt + ∫_{0}^{s} a cosh(s − t + \frac{1}{2})f(t)dt
+a sinh(−\frac{1}{2})f(s) − a sinh(\frac{1}{2})f(s)
= −f(s) + ∫_{0}^{1} G(s, t)f(t)dt

for almost all s ∈ [0, 1]. □

Definition 2.1.6 Because \frac{d}{ds}(∫_{0}^{1} G_{s}(s, t)f(t))dt = ∫_{0}^{1} G(s, t)f(t)dt − f(s) as we have seen in the lemma above, let us reserve the notation G_{ss}(s, t) to mean

G_{ss}(s, t) ≡ G(s, t) − δ(t − s)

where δ is the Dirac delta generalized function, i.e.,

∫_{−∞}^{+∞} δ(t − s)g(t)dt ≡ g(s)

in the distributional sense.

Thus,

\frac{d}{ds}(∫_{0}^{1} G_{s}(s, t)f(t)dt) = ∫_{0}^{1} G_{ss}(s, t)f(t)dt

≡ ∫_{0}^{1} G(s, t)f(t)dt + f(s).

Lemma 2.1.7 For k, f ∈ H(𝔤), the unique solution, u ∈ H , to the following equation,

(u, k) = ∫_{0}^{1}{⟨u′, k′⟩ + ⟨u, k⟩}ds = ∫_{0}^{1}⟨f, k⟩ds  (2.7)

is

u(s) = ∫_{0}^{1} G(s, t)f(t) dt.

Proof: Using integration by parts for the first term, we rewrite (2.7) as

∫_{0}^{1}⟨−u′′ + u, k⟩ds = ∫_{0}^{1}⟨f, k⟩ds.

So we seek a solution to the second order linear differential equation

−u′′ + u = f

u(0) = u(1) f(0) = f(1)
u′(0) = u′(1) f′(0) = f′(1).

Checking that u = ∫_{0}^{1} G(s, t)f(t) dt is indeed our desired solution, we note that

by Lemmas 2.1.4 and 2.1.5

\frac{d}{ds}(u(s)) = ∫_{0}^{1} F (s, t)f(t)dt

and

\frac{d^{2}}{ds^{2}} (u(s)) = \frac{d}{ds}(∫_{0}^{1} F (s, t)f(t)dt)
= ∫_{0}^{1} G_{ss}(s, t)f(t)dt
= −f(s) + ∫_{0}^{1} G(s, t)f(t)dt

It then follows that

−\frac{d^{2}}{ds^{2}} (u(s)) + u(s) = ∫_{0}^{1}\{−G_{ss}(s, t) + G(s, t)\}f(t)dt = f(s).

Finally, note that the boundary conditions are indeed satisfied by u:

u(0) = ∫_{0}^{1} a cosh(−t + \frac{1}{2})f(t)dt = u(1)

u′(0) = ∫_{0}^{1} a sinh(−t + \frac{1}{2})f(t)dt = u′(1). □

(Proof of Proposition 2.1.3 continued) Using the above lemma, we can solve

(ad_{h}^{∗}l, k) = ∫_{0}^{1}{⟨(ad_{h}^{∗}l)′, k′⟩ + ⟨ad_{h}^{∗}l, k⟩}ds
= ∫_{0}^{1}⟨[l, h] + [l′, h′] + [h, l′]′, k⟩ds

to get

ad_{h}^{∗}l(s) = ∫_{0}^{1} G(s, t)([l, h] + [l′, h′] + [h, l′]′)(t)dt. □

Returning to equation (2.1), we can now determine ∇h̃l̃:

∇h̃l̃ = −\frac{1}{2}\{[l, h] + ad_{h}^{∗}l + ad_{l}^{∗}h\}~
= −\frac{1}{2}\{[l, h] + ∫_{0}^{1} G(s, t)([h, l′]′ + [l, h′]′)(t)dt\}~.

Let us define a derivative, also called ∇, on H in the obvious way:

∇_{h}l ≡ ∇h̃l̃|_{e}

where e is the identity of L(G). So

∇_{h}l(s) = −\frac{1}{2}\{[l, h](s) + ∫_{0}^{1} G(s, t)([h, l′]′ + [l, h′]′)(t)dt\}.  (2.8)

We would like to express (2.8) in a form that has no second derivatives. Noting

that G(s, t) is continuous with respect to t for all s ∈ [0, 1], and that

G(s, 0) = a cosh(s − \frac{1}{2}) = G(s, 1)

we may use integration by parts in (2.8):

∇_{h}l(s) = −\frac{1}{2}\{[l, h](s) + ∫_{0}^{1} G(s, t)([h, l′] + [l, h′])′(t)dt\}
= −\frac{1}{2}\{[l, h](s) + ∫_{0}^{1} G_{s}(s, t)([h, l′] + [l, h′])(t)dt\}
= −\frac{1}{2}\{[l, h](s) + ∫_{0}^{1} F (s, t)([h, l′] + [l, h′])(t)dt\}
= −\frac{1}{2}\{[l, h](s) + ∫_{0}^{1} F (s, t)[h, l′](t)dt + ∫_{0}^{1} F (s, t)[l, h′](t)dt\}. (2.9)

Recall that we are assuming [h, l′] and [l, h′] are both in H(𝔤).

By the continuity of G, the above integration by parts is straightforward. Before

determining a final expression for ∇_{h}l, we need to verify a number of other integrationby-parts formulas involving the Green’s function, G.

Proposition 2.1.8 Using the notation G_{ss}(s, t) from Definition 2.1.6, for h, k ∈ H(𝔤),

∫_{0}^{1}⟨∫_{0}^{1} G_{s}(s, t)h(t)dt, k′(s)⟩ds = ∫_{0}^{1}⟨∫_{0}^{1} G_{ss}(s, t)h(t)dt, k(s)⟩ds.

Proof: Applying integration by parts and Lemma 2.1.5 gives

∫_{0}^{1}⟨∫_{0}^{1} G_{s}(s, t)h(t)dt, k′(s)⟩ds = ∫_{0}^{1}⟨\frac{d}{ds}(∫_{0}^{1} G_{s}(s, t)h(t)dt), k(s)⟩ds
= ∫_{0}^{1}⟨∫_{0}^{1} G_{ss}(s, t)h(t)dt, k(s)⟩ds. □

Proposition 2.1.9 For f ∈ H(R) or f ∈ H(𝔤),

∫_{0}^{1} G_{s}(s, t)f′(s)ds = ∫_{0}^{1} −G_{ss}(s, t)f(s)ds.

Proof: Using integration by parts and the definition of G_{s}, ∫_{0}^{1} G_{s}(s, t)f′(s)ds

= ∫_{0}^{t} a sinh(s − t + \frac{1}{2})f′(s)ds + ∫_{t}^{1} a sinh(s − t − \frac{1}{2})f′(s)ds
= a sinh(t − t + \frac{1}{2})f(t) − a sinh(0 − t + \frac{1}{2})f(0) + ∫_{0}^{t} a cosh(s − t + \frac{1}{2})f(s)ds

+a sinh(1 − t − \frac{1}{2})f(1) − a sinh(t − t + \frac{1}{2})f(t) + ∫_{t}^{1} a cosh(s − t − \frac{1}{2})f(s)ds

= 2a sinh(\frac{1}{2})f(t) + a sinh(−t + \frac{1}{2})(f(1) − f(0)) + ∫_{0}^{1} a cosh(r(s, t))f(s)ds
= f(t) + ∫_{0}^{1} G(s, t)f(s)ds = ∫_{0}^{1}{δ(t − s) + G(s, t)}f(s)ds = ∫_{0}^{1} −G_{ss}(s, t)f(s)ds.□

Proposition 2.1.10 For f ∈ H(R) or f ∈ H(𝔤),

∫_{0}^{1} G_{t}(s, t)f′(s)ds = ∫_{0}^{1} −G_{ss}(s, t)f(s)ds.

Proof:

∫_{0}^{1} G_{t}(s, t)f′(s)ds

= − ∫_{0}^{s} a sinh(s − t − \frac{1}{2})f′(t)dt + ∫_{s}^{1} a sinh(s − t + \frac{1}{2})f′(t)dt
= −a sinh(s − s − \frac{1}{2})f(s) + a sinh(s − 0 − \frac{1}{2})f(0) − ∫_{0}^{s} a cosh(s − t − \frac{1}{2})f(t)dt

−a sinh(s − 1 + \frac{1}{2})f(1) + a sinh(s − s + \frac{1}{2})f(s) − ∫_{s}^{1} a cosh(s − t + \frac{1}{2})f(t)dt

= 2a sinh(\frac{1}{2})f(s) − ∫_{0}^{1} G(s, t)f(t)dt = ∫_{0}^{1}\{−G(s, t) + δ(t − s)\}f(t)dt
= ∫_{0}^{1} −G_{ss}(s, t)f(t)dt.□

Proposition 2.1.11

∫_{0}^{1} G_{s}(s, t_{1})G_{s}(s, t_{2})ds = ∫_{0}^{1} −G_{ss}(s, t_{1})G(s, t_{2})ds.

Proof: Assume first that t_{1} ≤ t_{2}. Then, recalling that

G_{s}(s, t) = \{ \begin{matrix}a sinh(s − t − \frac{1}{2}) if 0 ≤ t < s \\ a sinh(s − t + \frac{1}{2}) if s ≤ t ≤ 1 \end{matrix}

where a = _{\frac{1}{2 sinh( \frac{1}{2} )}}, we have the following:

∫_{0}^{1} G_{s}(s, t_{1})G_{s}(s, t_{2})ds

= ∫_{0}^{t_{1}} a sinh(s − t_{1} + \frac{1}{2})a sinh(s − t_{2} + \frac{1}{2})ds

+ ∫_{t_{1}}^{t_{2}} a sinh(s − t_{1} − \frac{1}{2})a sinh(s − t_{2} + \frac{1}{2})ds

∫_{t_{2}}^{1} a sinh(s − t_{1} − \frac{1}{2})a sinh(s − t_{2} − \frac{1}{2})ds

= a sinh(t_{1} − t_{1} + \frac{1}{2})a cosh(t_{1} − t_{2} + \frac{1}{2}) − a sinh(−t_{1} + \frac{1}{2})a cosh(−t_{2} + \frac{1}{2})

− ∫_{0}^{t_{1}} a cosh(s − t_{1} + \frac{1}{2})a cosh(s − t_{2} + \frac{1}{2})ds

+a sinh(t_{2} − t_{1} − \frac{1}{2})a cosh(t_{2} − t_{2} + \frac{1}{2}) − a sinh(t_{1} − t_{1} + \frac{1}{2})a cosh(t_{1} − t_{2} + \frac{1}{2})

− ∫_{t_{1}}^{t_{1}} a cosh(s − t_{1} − \frac{1}{2})a cosh(s − t_{2} + \frac{1}{2})ds

+a sinh(1 − t_{1} − \frac{1}{2})a cosh(1 − t_{2} − \frac{1}{2}) − a sinh(t_{2} − t_{1} − \frac{1}{2})a cosh(t_{2} − t_{2} − \frac{1}{2})

− ∫_{t_{1}}^{1} a cosh(s − t_{1} − \frac{1}{2})a cosh(s − t_{2} − \frac{1}{2})ds.

After cancelling like terms, we get

∫_{0}^{1} G_{s}(s, t_{1})G_{s}(s, t_{2})ds = 2a sinh(\frac{1}{2})a cosh(t_{1} − t_{2} + \frac{1}{2}) − ∫_{0}^{1} G(s, t_{1})G(s, t_{2})ds
= a cosh(t_{1} − t_{2} + \frac{1}{2}) − ∫_{0}^{1} G(s, t_{1})G(s, t_{2})ds.

Similarly, if t_{2} < t_{1},

∫_{0}^{1} G_{s}(s, t_{1})G_{s}(s, t_{2})ds = 2a sinh(\frac{1}{2})a cosh(t_{2} − t_{1} + \frac{1}{2}) − ∫_{0}^{1} G(s, t_{2})G(s, t_{1})ds
= a cosh(t_{1} − t_{2} − \frac{1}{2}) − ∫_{0}^{1} G(s, t_{1})G(s, t_{2})ds.

Thus, for all (t_{1}, t_{2}) ∈ [0, 1]^{2},

∫_{0}^{1} G_{s}(s, t_{1})G_{s}(s, t_{2})ds = G(t_{1}, t_{2}) − ∫_{0}^{1} G(s, t_{1})G(s, t_{2})ds
= ∫_{0}^{1} −G_{ss}(s, t_{1})G(s, t_{2})ds.□

Proposition 2.1.12

∫_{0}^{1} G(s, t_{1})G_{s}(s, t_{2})ds = − ∫_{0}^{1} G_{s}(s, t_{1})G(s, t_{2})ds.

Proof: This follows from normal integration by parts.□

Returning to our expression for ∇_{h}l in (2.9), we may use Proposition 2.1.9 with

the antisymmetry of F to get

∫_{0}^{1} F (s, t)[l, h′]dt = ∫_{0}^{1} F (s, t)([l, h]′ − [l′, h])(t)dt
= −[l, h](s) − ∫_{0}^{1} G(s, t)[h, l](t)dt + ∫_{0}^{1} F (s, t)[h, l′](t)dt.(2.10)

Thus, (2.9) becomes

∇_{h}l(s) = −\frac{1}{2}\{[l, h](s) + ∫_{0}^{1} F (s, t)[h, l′](t)dt − [l, h](s)
− ∫_{0}^{1} G(s, t)[h, l](t)dt + ∫_{0}^{1} F (s, t)[h, l′](t)dt}
= − ∫_{0}^{1} F (s, t)[h, l′](t)dt + \frac{1}{2} ∫_{0}^{1} G(s, t)[h, l](t)dt.

Since this expression does not involve [h, l′]′ or [l, h′]′, we may take this to be our definition of ∇_{h}l for general h and l:

Definition 2.1.13 For h, l ∈ H(𝔤 ), let

∇_{h}l(s) = − ∫_{0}^{1} F (s, t)[h, l′](t)dt + \frac{1}{2} ∫_{0}^{1} G(s, t)[h, l](t)dt.

Before proceeding, it is important to verify that ∇_{h}l is actually an element of

the Cameron-Martin Hilbert space H(𝔤).

Proposition 2.1.14 If h, l ∈ H(𝔤), then ∇_{h}l ∈ H(𝔤). Forthermore, for almost all s ∈ [0, 1],

(∇_{h}l)′(s) = [h, l′](s) − ∫_{0}^{1} G(s, t)[h, l′](t)dt + \frac{1}{2} ∫_{0}^{1} F (s, t)[h, l](t)dt. (2.11)

Proof: The fact that ∇_{h}l is absolutely continuous follows directly from defi-

nition 2.1.13 and Lemmas 2.1.4 and 2.1.5. We may also obtain (2.11) from Lemmas 2.1.4 and 2.1.5. It is easy to check that (∇_{h}l, ∇_{h}l)_{H(𝔤)} < ∞ by use of Cauchy-Schwartz inequality and the fact that G(s, t) and F (s, t) are bounded for (s, t) ∈ [0, 1]^{2}. □

Proposition 2.1.15 ∇, as defined in Definition 2.1.13, is the “Levi-Civita covariant derivative” on H(𝔤 ).

Proof: We need to check directly that ∇ is torsion free and metric compatible..

Using Equation (2.10), we have

∇_{h}l − ∇_{l}h = − ∫_{0}^{1} F (s, t)[h, l′](t)dt + \frac{1}{2} ∫_{0}^{1} G(s, t)[h, l](t)dt

+ ∫_{0}^{1} F (s, t)[l, h′](t)dt − \frac{1}{2} ∫_{0}^{1} G(s, t)[l, h](t)dt

= − ∫_{0}^{1} F (s, t)[h, l′](t)dt + ∫_{0}^{1} G(s, t)[h, l](t)dt

−[l, h](s) − ∫_{0}^{1} G(s, t)[h, l](t)dt + ∫_{0}^{1} F (s, t)[h, l′](t)dt

= −[l, h](s) = [h, l](s)

proving ∇ is torsion free.

For metric compatibility we want to show that

(∇_{h}l, k) + (l, ∇_{h}k) = 0 ∀h, l, k ∈ H.

Note the following:

(∇_{h}l, k) + (l, ∇_{h}k)

= ∫_{0}^{1}{⟨(∇_{h}l)′, k′⟩ + ⟨l′, (∇_{h}k)′⟩ + ⟨∇_{h}l, k⟩ + ⟨l, ∇_{h}k⟩}ds
= ∫_{0}^{1}\{⟨[h, l′](s) − ∫_{0}^{1} G(s, t)[h, l′](t)dt + \frac{1}{2} ∫_{0}^{1} F (s, t)[h, l](t)dt, k′⟩

+⟨l′(s), [h, k′](s) − ∫_{0}^{1} G(s, t)[h, k′](t)dt + \frac{1}{2} ∫_{0}^{1} F (s, t)[h, k](t)dt⟩

+⟨− ∫_{0}^{1} F (s, t)[h, l′](t)dt + \frac{1}{2} ∫_{0}^{1} G(s, t)[h, l](t)dt, k⟩

+⟨l(s), − ∫_{0}^{1} F (s, t)[h, k′](t)dt + \frac{1}{2} ∫_{0}^{1} G(s, t)[h, k](t)dt⟩\}ds

= ∫_{0}^{1}{⟨[h, l′], k′⟩ + ⟨l′, [h, k′]⟩}ds

− ∫_{0}^{1}⟨− ∫_{0}^{1} F (s, t)[h, l′](t)dt + \frac{1}{2}(−[h, l](s) + ∫_{0}^{1} G(s, t)[h, l](t)dt), k⟩ds

− ∫_{0}^{1}⟨l(s), − ∫_{0}^{1} F (s, t)[h, k′](t)dt + \frac{1}{2}(−[h, k](s) + ∫_{0}^{1} G(s, t)[h, k](t)dt)⟩ds

+ ∫_{0}^{1}⟨− ∫_{0}^{1} F (s, t)[h, l′](t)dt + \frac{1}{2} ∫_{0}^{1} G(s, t)[h, l](t)dt, k⟩ds

+ ∫_{0}^{1}⟨l(s), − ∫_{0}^{1} F (s, t)[h, k′](t)dt + \frac{1}{2} ∫_{0}^{1} G(s, t)[h, k](t)dt⟩ds

= ∫_{0}^{1}{⟨[h, l′], k′⟩ − ⟨[h, l′], k′⟩}ds

− ∫_{0}^{1} \frac{1}{2}\{⟨−[h, l], k⟩ + ⟨l, −[h, k]⟩\}ds

= − ∫_{0}^{1} \frac{1}{2}\{−⟨[h, l], k⟩ + ⟨[h, l], k⟩\}ds
= 0. □

Note that since G_{s} = −G_{t}, we may write

∇_{h}l(s) = ∫_{0}^{1} G_{t}(s, t)[h, l′](t)dt + \frac{1}{2} ∫_{0}^{1} G(s, t)[h, l](t)dt.

By comparison, the covariant derivative, D, for the Lie algebra of the pinned loop group, as found in Driver [5], is

D_{h}l(s) = ∫_{0}^{s}[h, l′]dt − s ∫_{0}^{1}[h, l′]dt = ∫_{0}^{1}{θ(s − t) − s}[h, l′](t)dt.

The Green’s function for that case is G_{0}(s, t) ≡ s ∧ t − st. So

G_{0t}(s, t) = \{ \begin{matrix}1 − s & t ≤ s \\ −s & s < t. \end{matrix}
= (1 − s)1_{t≤s} − s1_{t>s}
= 1_{t≤s} − s
= θ(s − t) − s.

Thus,

D_{h}l(s) = ∫_{0}^{1} G_{0t}(s, t)[h, l′](t)dt.

2.2 Curvature

Recall that the covariant derivative turned out to be

∇_{h}l(s) = − ∫_{0}^{1} F (s, t)[h, l′](t)dt + \frac{1}{2} ∫_{0}^{1} G(s, t)[h, l](t)dt.

We wish to calculate the curvature tensor

R⟨h, k⟩l = [∇_{h}, ∇_{k}]l − ∇_{[h,k]}l.

for h, k, l ∈ H. First note:

∇_{h}∇_{k}l(x)

= ∫_{0}^{1} −F (x, s)[h, (∇_{k}l)′](s)ds + \frac{1}{2} ∫_{0}^{1} G(x, s)[h, ∇_{k}l](s)ds
= ∫_{0}^{1} −F (x, s)[h(s), [k, l′](s) − ∫_{0}^{1} G(s, t)[k, l′](t)dt + \frac{1}{2} ∫_{0}^{1} F (s, t)[k, l](t)dt]ds

+\frac{1}{2} ∫_{0}^{1} G(x, s)[h(s), − ∫_{0}^{1} F (s, t)[k, l′](t)dt + \frac{1}{2} ∫_{0}^{1} G(s, t)[k, l](t)dt]ds.

So

[∇_{h}, ∇_{k}]l(x)

= ∫_{0}^{1} −F (x, s)[h, [k, l′]]ds − ∫_{0}^{1} −F (x, s)[k, [h, l′]]ds

+ ∫_{0}^{1} −F (x, s)[h(s), − ∫_{0}^{1} G(s, t)[k, l′](t)dt + \frac{1}{2} ∫_{0}^{1} F (s, t)[k, l](t)dt]ds

− ∫_{0}^{1} −F (x, s)[k(s), − ∫_{0}^{1} G(s, t)[h, l′](t)dt + \frac{1}{2} ∫_{0}^{1} F (s, t)[h, l](t)dt]ds

+\frac{1}{2} ∫_{0}^{1} G(x, s)[h(s), − ∫_{0}^{1} F (s, t)[k, l′](t)dt + \frac{1}{2} ∫_{0}^{1} G(s, t)[k, l](t)dt]ds

−\frac{1}{2} ∫_{0}^{1} G(x, s)[k(s), − ∫_{0}^{1} F (s, t)[h, l′](t)dt + \frac{1}{2} ∫_{0}^{1} G(s, t)[h, l](t)dt]ds

= ∫_{0}^{1} F (x, s)[l′, [h, k]](s)ds

+ ∫_{0}^{1} ∫_{0}^{1}\{F (x, s)G(s, t) − \frac{1}{2}G(x, s)F (s, t)\}[h(s), [k, l′](t)]dtds

+ ∫_{0}^{1} ∫_{0}^{1}\{\frac{1}{2}F (x, s)F (s, t) − \frac{1}{4}G(x, s)G(s, t)\}[h(s), [l, k](t)]dtds

− ∫_{0}^{1} ∫_{0}^{1}\{F (x, s)G(s, t) − \frac{1}{2}G(x, s)F (s, t)\}[k(s), [h, l′](t)]dtds

− ∫_{0}^{1} ∫_{0}^{1}\{\frac{1}{2}F (x, s)F (s, t) − \frac{1}{4}G(x, s)G(s, t)\}[k(s), [l, h](t)]dtds

= ∫_{0}^{1} F (x, s)[l′, [h, k]](s)ds

+ ∫_{0}^{1} ∫_{0}^{1} M(x, s, t)([h(s), [k, l′](t)] − [k(s), [h, l′](t)])dtds

+\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} N(x, s, t)([h(s), [l, k](t)] − [k(s), [l, h](t)])dtds

where we have used the Jacobi identity for the first term in the second equation above, and we define M and N by

M(x, s, t) ≡ F (x, s)G(s, t) − \frac{1}{2}G(x, s)F (s, t)  (2.12)
N(x, s, t) ≡ F (x, s)F (s, t) − \frac{1}{2}G(x, s)G(s, t).  (2.13)

Finally, noting that

∇_{[h,k]}l(x) = − ∫_{0}^{1} F (x, s)[[h, k], l′](s)ds + \frac{1}{2} ∫_{0}^{1} G(x, s)[[h, k], l](s)ds

we get the following theorem. Theorem 2.2.1 For h, k, l ∈ H(𝔤), the curvature tensor can be written as

(R⟨h, k⟩l)(x) = ∫_{0}^{1} ∫_{0}^{1} M(x, s, t)([h(s), [k, l′](t)] − [k(s), [h, l′](t)])dtds

+\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} N(x, s, t)([h(s), [l, k](t)] − [k(s), [l, h](t)])dtds

+\frac{1}{2} ∫_{0}^{1} G(x, s)[l, [h, k]](s)ds,

where M and N are defined in (2.12) and (2.13).

2.3 Ricci Tensor

If h, k ∈ H and k has the property that [k′, k] = 0, then

(R⟨h, k⟩k)(x) = ∫_{0}^{1} ∫_{0}^{1} M(x, s, t)([h(s), [k, k′](t)] − [k(s), [h, k′](t)])dtds

+\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} N(x, s, t)([h(s), [k, k](t)] − [k(s), [k, h](t)])dtds

+\frac{1}{2} ∫_{0}^{1} G(x, s)[k, [h, k]](s)ds

= ∫_{0}^{1} ∫_{0}^{1} M(x, s, t)[k(s), [k′, h](t)]dtds

−\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} N(x, s, t)[k(s), [k, h](t)]dtds

+\frac{1}{2} ∫_{0}^{1} G(x, s)[k, [h, k]](s)ds.  (2.14)

In anticipation of the Ricci tensor, we wish now to compute (R⟨h, k⟩k, h). For

what follows, the x, s and t in M(x, s, t) and N(x, s, t) will be suppressed. Also, let us keep in mind that our Green’s function, G(x, s), has the following properties, as discussed in Definition 2.1.6:

−G_{xx}(x, s) + G(x, s) = δ(s − x)
F (x, s) = G_{x}(x, s)
G_{s}(x, s) = −G_{x}(x, s).

First consider

(∫_{0}^{1} G(·, s)[k, [h, k]](s)ds, h(·))

= ∫_{0}^{1}{⟨∫_{0}^{1} G_{x}(x, s)[k, [h, k]](s)ds, h′(x)⟩ + ⟨∫_{0}^{1} G(x, s)[k, [h, k]](s)ds, h(x)⟩}dx
= ∫_{0}^{1}{−⟨∫_{0}^{1} G_{xx}(x, s)[k, [h, k]](s)ds, h(x)⟩ + ⟨∫_{0}^{1} G(x, s)[k, [h, k]](s)ds, h(x)⟩}dx
= ∫_{0}^{1}⟨∫_{0}^{1}(−G_{xx}(x, s) + G(x, s))[k, [h, k]](s)ds, h(x)⟩dx
= ∫_{0}^{1}⟨∫_{0}^{1} δ(s − x)[k, [h, k]](s)ds, h(x)⟩dx
= ∫_{0}^{1}⟨[k, [h, k]](x), h(x)⟩dx.

Thus,

(∫_{0}^{1} G(·, s)[k, [h, k]](s)ds, h(·)) = ∫_{0}^{1}⟨[k, h](x), [k, h](x)⟩dx.  (2.15)

Next, consider (∫_{0}^{1} ∫_{0}^{1} M(·, s, t)[k(s), [k′, h](t)]dtds, h(·))

= ∫_{0}^{1}{⟨∫_{0}^{1} ∫_{0}^{1} M_{x}[k(s), [k′, h](t)]dtds, h′(x)⟩

+⟨∫_{0}^{1} ∫_{0}^{1} M[k(s), [k′, h](t)]dtds, h(x)⟩}dx

= ∫_{0}^{1}\{⟨∫_{0}^{1} ∫_{0}^{1}\{G_{xx}(x, s)G(s, t) − \frac{1}{2}G_{x}(x, s)F (s, t)\}[k(s), [k′, h](t)]dtds, h′(x)⟩

+⟨∫_{0}^{1} ∫_{0}^{1} M[k(s), [k′, h](t)]dtds, h(x)⟩}dx

= ∫_{0}^{1}{⟨∫_{0}^{1} ∫_{0}^{1}{(G(x, s) − δ(s − x))G(s, t)

−\frac{1}{2}G_{x}(x, s)F (s, t)\}[k(s), [k′, h](t)]dtds, h′(x)⟩

+⟨∫_{0}^{1} ∫_{0}^{1} M[k(s), [k′, h](t)]dtds, h(x)⟩}dx.  (2.16)

Let us calculate different parts of (2.16) separately:

a) ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1} −δ(s − x)G(s, t)[k(s), [k′, h](t)]dtds, h′(x)⟩dx

= ∫_{0}^{1}⟨∫_{0}^{1}[∫_{0}^{1} −δ(s − x)G(s, t)k(s)ds, [k′, h](t)]dt, h′(x)⟩dx
= ∫_{0}^{1}⟨∫_{0}^{1} −[G(x, t)k(x), [k′, h](t)]dt, h′(x)⟩dx
= ∫_{0}^{1}⟨∫_{0}^{1}[G_{x}(x, t)k(x) + G(x, t)k′(x), [k′, h](t)]dt, h(x)⟩dx
= ∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k(x), [k′, h](t)], h(x)⟩dtdx

+ ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k′(x), [k′, h](t)], h(x)⟩dtdx b) ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1}{G(x, s)G(s, t)

−\frac{1}{2}G_{x}(x, s)F (s, t)\}[k(s), [k′, h](t)]dtds, h′(x)⟩dx

= − ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1}\{G_{x}(x, s)G(s, t) − \frac{1}{2}G_{xx}(x, s)F (s, t)\}[k(s), [k′, h](t)]dtds, h(x)⟩dx
= − ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1}\{G_{x}(x, s)G(s, t) − \frac{1}{2}(G(x, s) − δ(s − x))F (s, t)\}

·[k(s), [k′, h](t)]dtds, h(x)⟩dx

= − ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1}\{F (x, s)G(s, t) − \frac{1}{2}G(x, s)F (s, t)\}[k(s), [k′, h](t)]dtds, h(x)⟩dx

− ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1} \frac{1}{2}δ(s − x)F (s, t)[k(s), [k′, h](t)]dtds, h(x)⟩dx

= − ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1} M[k(s), [k′, h](t)]dtds, h(x)⟩dx

− ∫_{0}^{1}⟨∫_{0}^{1} \frac{1}{2}F (x, t)[k(x), [k′, h](t)]dt, h(x)⟩dx

= − ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1} M[k(s), [k′, h](t)]dtds, h(x)⟩dx

−\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k(x), [k′, h](t)], h(x)⟩dtdx

Using these calculations in (2.16) we get

(∫_{0}^{1} ∫_{0}^{1} M(·, s, t)[k(s), [k′, h](t)]dtds, h(·))

= ∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k(x), [k′, h](t)], h(x)⟩dtdx

+ ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k′(x), [k′, h](t)], h(x)⟩dtdx − ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1} M[k(s), [k′, h](t)]dtds, h(x)⟩dx

−\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k(x), [k′, h](t)], h(x)⟩dtdx

+ ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1} M[k(s), [k′, h](t)]dtds, h(x)⟩dx

= ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k′(x), [k′, h](t)], h(x)⟩dtdx

+\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k(x), [k′, h](t)], h(x)⟩dtdx,

and thus

(∫_{0}^{1} ∫_{0}^{1} M(·, s, t)[k(s), [k′, h](t)]dtds, h(·))
= − ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k′, h](t), [k′, h](x)⟩dtdx
−\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k′, h](t), [k, h](x)⟩dtdx.  (2.17)

Finally, we compute (∫_{0}^{1} ∫_{0}^{1} N[k(s), [k, h](t)]dtds, h(x))

= ∫_{0}^{1}{⟨∫_{0}^{1} ∫_{0}^{1} N_{x}[k(s), [k, h](t)]dtds, h′(x)⟩

+⟨∫_{0}^{1} ∫_{0}^{1} N[k(s), [k, h](t)]dtds, h(x)⟩}dx

= ∫_{0}^{1}\{⟨∫_{0}^{1} ∫_{0}^{1}\{G_{xx}(x, s)F (s, t) − \frac{1}{2}G_{x}(x, s)G(s, t)\}[k(s), [k, h](t)]dtds, h′(x)⟩

+⟨∫_{0}^{1} ∫_{0}^{1} N[k(s), [k, h](t)]dtds, h(x)⟩}dx

= ∫_{0}^{1}{⟨∫_{0}^{1} ∫_{0}^{1}{(G(x, s) − δ(s − x))F (s, t)

−\frac{1}{2}G_{x}(x, s)G(s, t)\}[k(s), [k, h](t)]dtds, h′(x)⟩

+⟨∫_{0}^{1} ∫_{0}^{1} N[k(s), [k, h](t)]dtds, h(x)⟩}dx  (2.18)

Let us calculate different parts of (2.18) separately:

a) ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1} −δ(s − x)F (s, t)[k(s), [k, h](t)]dtds, h′(x)⟩dx

= ∫_{0}^{1}⟨∫_{0}^{1}[∫_{0}^{1} −δ(s − x)F (s, t)k(s)ds, [k, h](t)]dt, h′(x)⟩dx
= ∫_{0}^{1} −⟨∫_{0}^{1}[F (x, t)k(x), [k, h](t)]dt, h′(x)⟩dx
= ∫_{0}^{1} −⟨[k(x), ∫_{0}^{1} F (x, t)[k, h](t)dt], h′(x)⟩dx
= ∫_{0}^{1}⟨[k′(x), ∫_{0}^{1} F (x, t)[k, h](t)dt] + [k(x), ∫_{0}^{1} G_{xx}(x, t)[k, h](t)dt], h(x)⟩dx
= ∫_{0}^{1}⟨[k′(x), ∫_{0}^{1} F (x, t)[k, h](t)dt] + [k(x), ∫_{0}^{1}{G(x, t) − δ(t − x)}[k, h](t)dt], h(x)⟩dx

= ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k(x), [k, h](t)], h(x)⟩dtdx

+ ∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k′(x), [k, h](t)], h(x)⟩dtdx

+ ∫_{0}^{1}⟨[k, h](t), [k, h](t)⟩dt

b) ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1}\{(G(x, s)F (s, t) − \frac{1}{2}G_{x}(x, s)G(s, t)\}[k(s), [k, h](t)]dtds, h′(x)⟩dx

= − ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1}\{(G_{x}(x, s)F (s, t) − \frac{1}{2}G_{xx}(x, s)G(s, t)\}[k(s), [k, h](t)]dtds, h(x)⟩dx
= − ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1}{(G_{x}(x, s)F (s, t)

−\frac{1}{2}(G(x, s) − δ(s − x))G(s, t)\}[k(s), [k, h](t)]dtds, h(x)⟩dx

= − ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1} N[k(s), [k, h](t)]dtds, h(x)⟩dx

− ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1} \frac{1}{2}δ(s − x)G(s, t)[k(s), [k, h](t)]dtds, h(x)⟩dx

= − ∫_{0}^{1}⟨∫_{0}^{1} ∫_{0}^{1} N[k(s), [k, h](t)]dtds, h(x)⟩dx

−\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k(x), [k, h](t)], h(x)⟩dtdx

Using these calculations in (2.18) we get

(∫_{0}^{1} ∫_{0}^{1} N[k(s), [k, h](t)]dtds, h(x))

= ∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k′(x), [k, h](t)], h(x)⟩dtdx

+\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k(x), [k, h](t)], h(x)⟩dtdx

+ ∫_{0}^{1}⟨[k, h](t), [k, h](t)⟩dt,

and so

(∫_{0}^{1} ∫_{0}^{1} N[k(s), [k, h](t)]dtds, h(x)) = − ∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k, h](t), [k′, h](x)⟩dtdx

−\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k, h](t), [k, h](x)⟩dtdx

+ ∫_{0}^{1}⟨[k, h](t), [k, h](t)⟩dt.  (2.19)

Now we are ready to determine (R⟨h, k⟩k, h). Putting Equations (2.15), (2.17), and (2.19) together gives

(R⟨h, k⟩k, h) = (∫_{0}^{1} ∫_{0}^{1} M[k(s), [k′, h](t)]dtds, h)

−\frac{1}{2}(∫_{0}^{1} ∫_{0}^{1} N(x, s, t)[k(s), [k, h](t)]dtds, h)

+\frac{1}{2}(∫_{0}^{1} G(x, s)[k, [h, k]]ds, h)

= − ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k′, h](t), [k′, h](x)⟩dtdx

−\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k′, h](t), [k, h](x)⟩dtdx

+\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k, h](t), [k′, h](x)⟩dtdx

+\frac{1}{4} ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k, h](t), [k, h](x)⟩dtdx

−\frac{1}{2} ∫_{0}^{1}⟨[k, h](t), [k, h](t)⟩dt

+\frac{1}{2} ∫_{0}^{1}⟨[k, h](x), [k, h](x)⟩dx

= − ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k′, h](t), [k′, h](x)⟩dtdx

+\frac{1}{4} ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k, h](t), [k, h](x)⟩dtdx

−\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k′, h](t), [k, h](x)⟩dtdx

+\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} F (t, x)⟨[k, h](x), [k′, h](t)⟩dtdx.

We have thus proved the following theorem. Theorem 2.3.1

(R⟨h, k⟩k, h) = − ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k′, h](x), [k′, h](t)⟩dtdx

+\frac{1}{4} ∫_{0}^{1} ∫_{0}^{1} G(x, t)⟨[k, h](x), [k, h](t)⟩dtdx

− ∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k, h](x), [k′, h](t)⟩dtdx.  (2.20)

Recalling that F (x, t) = G_{x}(x, t), we may use integration by parts to see that

∫_{0}^{1} ∫_{0}^{1} F (x, t)⟨[k, h](x), [k′, h](t)⟩dtdx

= ∫_{0}^{1} ∫_{0}^{1} −G(x, t)⟨[k′, h](x) + [k, h′](x), [k′, h](t)⟩dtdx.

Thus, (2.20) can be written in the alternative form

(R⟨h, k⟩k, h) = ∫_{0}^{1} ∫_{0}^{1} G(x, t)\{⟨[k, h′](x), [k′, h](t)⟩ + \frac{1}{4}⟨[k, h](x), [k, h](t)⟩\}dtdx.

This verifies the following symmetry property of the curvature tensor. Corollary 2.3.2 For h, k ∈ H(𝔤),

(R⟨h, k⟩k, h) = (R⟨k, h⟩h, k).

Before determining a suitable expression for the Ricci tensor, it will be useful

to first point out some properties of our recurring Green’s function, G. Definition 2.3.3 Let H(R) be the space of absolutely continuous functions a : [0, 1] → R with a(0) = a(1). Definition 2.3.4 Let S be the orthonormal basis of H(g) (the Lie algebra of our free loop group L(G)), with the form S = {aA : a ∈ h_{0}, A ∈ g_{0}}, where g_{0} is an orthonormal basis of 𝔤, and h_{0} is an orthonormal basis of H(R).

Throughout this paper we will take “orthonormal basis” to mean a Hilbert

basis (i.e., total and orthonormal in the given Hilbert space). Lemma 2.3.5 For all k ∈ S, [k′, k] = 0. Proof: Since k is of the form k(s) = a(s)A, a ∈ H(R), A ∈ g_{0}, then

[k′, k](s) = [a′(s)A, a(s)A] = a′(s)a(s)[A, A] = 0.

Q.E.D.

We will call such an S (i.e., with the property that [k, k′] = 0 for all k ∈ S) a

“good” basis of H(𝔤). Lemma 2.3.6 The Green’s function G(s, t) is the reproducing Kernel for H(R):

(G(s, ·), a) = a(s) ∀a ∈ H(R).

Proof: The inner product, (·, ·), that we are assuming gives

(G(s, ·), a) = ∫_{0}^{1}{G_{t}(s, t)a′(t) + G(s, t)a(t)}dt
= G_{t}(s, 1)a(1) − G_{t}(s, 0)a(0) + ∫_{0}^{1}(−G_{ss}(s, t) + G(s, t))a(t)dt
= ∫_{0}^{1} δ(t − s)a(t)dt
= a(s).

Q.E.D.

The next lemma provides a convenient way of writing G(s, t) as a series.

Lemma 2.3.7 For any orthonormal basis h_{0} of H(R), ∑_{a∈h_{0}} a(s)a(t) = G(s, t) ∀s, t ∈ [0, 1] Proof: Since G(s, t) is the reproducing kernel for H(R), and h_{0} is an orthonormal basis, then

G(s, t) = (G(s, ·), G(·, t))
= ( ∑_{a∈h_{0}}(G(s, ·), a)a, ∑_{b∈h_{0}}(G(·, t), b)b)
= ∑_{a∈h_{0}}(G(s, ·), a)(G(·, t), a)
= ∑_{a∈h_{0}} a(s)a(t).

Note that convergence of the series ∑_{a∈h_{0}} a(s)a(t) is guaranteed by the fact that h_{0} is a Hilbert basis. □ Lemma 2.3.8 For any orthonormal basis h_{0} of H(R),

∑_{a∈h_{0}} |a(s)a(t)| ≤ \frac{e + 1}{2(e − 1)} ∀s, t ∈ [0, 1].

Thus, ∑_{a∈h_{0}} a(s)a(t) is absolutely convergent.

Proof: From the previous lemma, for s ∈ [0, 1]

∑_{a∈h_{0}} a^{2}(s) = G(s, s)
= a cosh(\frac{1}{2})
= \frac{1}{2 sinh(\frac{1}{2})} cosh(\frac{1}{2})
= \frac{1}{2} · \frac{e^{\frac{1}{2}} + e^{− \frac{1}{2}}}{e^{\frac{1}{2}} − e^{− \frac{1}{2}}}
= \frac{e + 1}{2(e − 1)}

Thus, using the Cauchy Schwartz inequality,

∑_{a∈h_{0}} |a(s)a(t)| ≤ \sqrt{∑_{a∈h_{0}} a^{2}(s)}\sqrt{∑_{a∈h_{0}} a^{2}(t)}
= \frac{e + 1}{2(e − 1)}. □

Definition 2.3.9 Let K⟨·, ·⟩ denote the following form on 𝔤:

K⟨A, B⟩ ≡ −tr(ad_{A}ad_{B})

= − ∑_{C∈g_{0}}⟨ad_{A}ad_{B}C, C⟩

= ∑_{C∈g_{0}}⟨ad_{B}C, ad_{A}C⟩.

One should note that −K⟨·, ·⟩ is traditionally known as the Killing form of 𝔤 (see p. 131 in Helgason [11]).

Using the above lemmas and definitions, the Ricci tensor can now be computed

as follows.

Ric⟨h, h⟩ = ∑_{k∈S}(R⟨h, k⟩k, h)

= ∑_{k∈S} ∫_{0}^{1} ∫_{0}^{1}{−G(x, t)⟨[k′, h](x), [k′, h](t)⟩

+\frac{1}{4}G(x, t)⟨[k, h](x), [k, h](t)⟩

−F (x, t)⟨[k, h](x), [k′, h](t)⟩}dtdx

= ∑_{a∈h_{0},A∈g_{0}} ∫_{0}^{1} ∫_{0}^{1}{−G(x, t)⟨[a′(x)A, h(x)], [a′(t)A, h(t)]⟩

+\frac{1}{4}G(x, t)⟨[a(x)A, h(x)], [a(t)A, h(t)]⟩

−F (x, t)⟨[a(x)A, h(x)], [a′(t)A, h(t)]⟩}dtdx

= ∑_{a∈h_{0},A∈g_{0}} ∫_{0}^{1} ∫_{0}^{1}{G_{x}(x, t)⟨[a(x)A, h(x)], [a′(t)A, h(t)]⟩

+G(x, t)⟨[a(x)A, h′(x)], [a′(t)A, h(t)]⟩

+\frac{1}{4}G(x, t)⟨[a(x)A, h(x)], [a(t)A, h(t)]⟩

−G_{x}(x, t)⟨[a(x)A, h(x)], [a′(t)A, h(t)]⟩}dtdx

= ∑_{a∈h_{0},A∈g_{0}} ∫_{0}^{1} ∫_{0}^{1}{G_{x}(x, t)⟨[a(x)A, h(x)], [a(t)A, h(t)]⟩

−G(x, t)⟨[a(x)A, h(x)], [a(t)A, h′(t)]⟩

+\frac{1}{4}G(x, t)⟨[a(x)A, h(x)], [a(t)A, h(t)]⟩\}dtdx

= ∫_{0}^{1} ∫_{0}^{1}{G_{x}(x, t)G(x, t)K⟨h′(x), h(t)⟩

−G^{2}(x, t)K⟨h′(x), h′(t)⟩ + \frac{1}{4}G^{2}(x, t)K⟨h(x), h(t)⟩\}dtdx

= ∫_{0}^{1} ∫_{0}^{1}{(−G_{x}(x, t)G(x, t) − G_{x}^{2}(x, t))K⟨h(x), h(t)⟩

+2G_{x}(x, t)G(x, t)K⟨h(x), h′(t)⟩ + \frac{1}{4}G^{2}(x, t)K⟨h(x), h(t)⟩\}dtdx

= ∫_{0}^{1} G(x, x)K⟨h(x), h(x)⟩dx

+ ∫_{0}^{1} ∫_{0}^{1}{−G^{2}(x, t) − G_{x}^{2}(x, t) + 2G_{xx}(x, t)G(x, t)

+2G_{x}(x, t)^{2} + \frac{1}{4}G^{2}(x, t)\}K⟨h(x), h(t)⟩dtdx

= ∫_{0}^{1} G(x, x)K⟨h(x), h(x)⟩dx − 2 ∫_{0}^{1} G(x, x)K⟨h(x), h(x)⟩dx

+ ∫_{0}^{1} ∫_{0}^{1}{−G^{2}(x, t) − G_{x}^{2}(x, t) + 2G^{2}(x, t)

+2G_{x}(x, t)^{2} + \frac{1}{4}G^{2}(x, t)\}K⟨h(x), h(t)⟩dtdx.

We have thus proved the following theorem.

Theorem 2.3.10 For all h ∈ H(𝔤),

Ric⟨h, h⟩ = ∫_{0}^{1} ∫_{0}^{1}\{\frac{5}{4}G^{2}(x, t) + F^{2}(x, t)\}K⟨h(x), h(t)⟩dtdx  (2.21)
− ∫_{0}^{1} G(x, x)K⟨h(x), h(x)⟩dx,

where, again, G and F are defined as in (2.4).

An important key to proving our logarithmic Sobolev inequality in Theorem

6.0.2 will be based on the fact the Ricci tensor is bounded below, which is what we next verify.

Theorem 2.3.11 There exists a constant C > 0 such that

Ric⟨h, h⟩ ≥ −C(h, h)  (2.22)

for all h ∈ H(𝔤).

Proof: By the definition of G and F, it is clear that there exists bounds, C_{1}

and C_{2}, on such that |G(x, t)| ≤ C_{1} and |F (x, t)| ≤ C_{2} for all (x, t) ∈ [0, 1]^{2}. It then follows that for all h ∈ H(𝔤),

|Ric⟨h, h⟩| ≤ ∫_{0}^{1} ∫_{0}^{1}\{\frac{5}{4}C_{1}^{2} + C_{2}^{2}\} |K⟨h(x), h(t)⟩| dtdx + ∫_{0}^{1} C_{1} |K⟨h(x), h(x)⟩| dx
≤ ∫_{0}^{1} ∫_{0}^{1}\{\frac{5}{4}C_{1}^{2} + C_{2}^{2}\}c |⟨h(x), h(t)⟩_{𝔤}| dtdx + ∫_{0}^{1} C_{1}c⟨h(x), h(x)⟩_{𝔤}dx
≤ \{\frac{5}{4}C_{1}^{2} + C_{2}^{2}\}c ∫_{0}^{1} ∥h(t)∥_{𝔤} dt ∫_{0}^{1} ∥h(x)∥_{𝔤} dx + C_{1}c ∫_{0}^{1} ∥h(x)∥_{𝔤}^{2} dx
≤ \{\frac{5}{4}C_{1}^{2} + C_{2}^{2}\}c(∫_{0}^{1} ∥h(x)∥_{𝔤} dx)^{2} + C_{1}c ∫_{0}^{1} ∥h(x)∥_{𝔤}^{2} dx
≤ \{\frac{5}{4}C_{1}^{2} + C_{2}^{2}\}c ∫_{0}^{1} ∥h(x)∥_{𝔤}^{2} dx + C_{1}c ∫_{0}^{1} ∥h(x)∥_{𝔤}^{2} dx
≤ \{\frac{5}{4}C_{1}^{2} + C_{2}^{2} + C_{1}\}c ∫_{0}^{1}\{∥h′(x)∥_{𝔤}^{2} + ∥h(x)∥_{𝔤}^{2}\}dx
= \{\frac{5}{4}C_{1}^{2} + C_{2}^{2} + C_{1}\}c(h, h),

where we have used Hölder’s and Cauchy Schwartz inequalities for the third inequality above and c is a constant depending on K. Letting C ≡ \{\frac{5}{4}C_{1}^{2} + C_{2}^{2} + C_{1}\}c we see that (2.22) holds. □

Finite Dimensional

Approximations

For the finite dimensional approximations to the space L(G), we consider, for

each partition P = {0 = s_{0} < s_{1} < ⋯ < s_{n−1} < 1}, the finite dimensional product Lie group G^{P} ≡ G^{n} = G × ⋯ × G. Recall the Green’s function G(s, t) = a cosh(r(s, t)) where a = _{\frac{1}{2 sinh( \frac{1}{2} )}} and

r(s, t) = \{ \begin{matrix}s − t − \frac{1}{2} & 0 ≤ t < s \\ s − t + \frac{1}{2} & s ≤ t ≤ 1. \end{matrix} .

We define a metric on the Lie algebra 𝔤^{P} by

(A⃗, B⃗)_{P} ≡ ∑_{i,j=1}^{n} Q_{ij}⟨A_{i}, B_{j}⟩,  (3.1)

where A⃗ = (A_{1}, …, A_{n}), B⃗ = (B_{1}, …, B_{n}) ∈ 𝔤^{P}, ⟨·, ·⟩ is the inner product on 𝔤, and Q is the inverse of the matrix {G(s_{i}, s_{j})}_{i,j=1}^{n}. We can then extend (·, ·)_{P} to a unique left-invariant Riemannian metric, which we will also denote by (·, ·)_{P}, on G^{P}.

Before continuing, we should verify that the matrix {G(s_{i}, s_{j})}_{i,j=1}^{n}is actually

invertible. Let π̃_{P} : H(𝔤) → 𝔤^{P} be the projection defined by

π̃_{P}(h) ≡ (h(s_{0}), …, h(s_{n−1}))

for h ∈ H(𝔤)..It is sufficient to show that it is positive definite. So assume that x = (x_{0}, x_{1}, …, x_{n−1}) ∈ R^{n}\{0}. Then, by Lemma 2.3.7,

∑_{i,j=0}^{n−1} G(s_{i}, s_{j})x_{i}x_{j} = ∑_{i,j=0}^{n−1} ∑_{a∈h_{0}} a(s_{i})a(s_{j})x_{i}x_{j}
= ∑_{a∈h_{0}}(∑_{i=0}^{n−1} a(s_{i})x_{i})(∑_{j=0}^{n−1} a(s_{j})x_{j})
= ∑_{a∈h_{0}}(x · (π̃_{P}(a)))^{2},

where h_{0} is an orthonormal basis of H(𝔤). The summation in the last equation will be greater than zero if we just choose the basis h_{0} to include an element a such that x · (π̃_{P}(a)) ≠ 0 (one may start with such an a and use the Gram-Schmidt process to construct the rest of h_{o}).

Next, define the subspace H_{P}(𝔤) ⊆ H(𝔤) by

H_{P}(𝔤) ≡ (Ker(π̃_{P}))^{⊥}.

Lemma 3.0.1 If P = {0 = s_{0} < s_{1} < ⋯ < s_{n−1} < 1} is a given partition of [0, 1], then the following are true:

a) If k, l ∈ H_{P}(𝔤) are such that k(s_{i}) = l(s_{i}) ∀ s_{i} ∈ P, then k ≡ l.

b) H_{P}(𝔤) = {k ∈ H(𝔤) : k(s) = ∑_{i=0}^{n−1} G(s, s_{i})A_{i}, A_{i} ∈ 𝔤}.
c) H_{P}(𝔤) = {k ∈ H(𝔤) : −k′′(s) + k(s) = 0 for all s ∈ [0, 1]\P}.

Proof: a)Suppose k, l ∈ H_{P}(𝔤) are such that k(s_{i}) = l(s_{i}) ∀ s_{i} ∈ P. Then

h ≡ k − l has the property that h(s_{i}) = 0 ∀ s_{i} ∈ P, and thus h ∈ Ker(π̃_{P}) ∩ H_{P}(𝔤) = Ker(π̃_{P}) ∩ (Ker(π̃_{P}))^{⊥} = {0}. Hence, h ≡ 0 and so k ≡ l.

b) Suppose k(s) = ∑_{i=0}^{n−1} G(s, s_{i})A_{i} where A_{i} ∈ 𝔤 for all i ∈ {0, 1, …, n − 1}.

Then for an arbitrary h ∈ Ker(π̃_{P}), the reproducing property of G gives

(h, k) = (h, ∑_{i=0}^{n−1} G(s, s_{i})A_{i})
= ∑_{i=0}^{n−1}⟨h(s_{i}), A_{i}⟩ = 0,

where, for the last equation, we have used the fact that h(s_{i}) = 0 for all i. Since h ∈ Ker(π̃_{P}) was arbitrary, we have shown that k ∈ H_{P}(𝔤) = (Ker(π̃_{P}))^{⊥}.

Now suppose that k ∈ H_{P}(𝔤). We wish to show that it must be of the form

k(s) = ∑_{i=0}^{n−1} G(s, s_{i})A_{i} where A_{i} ∈ 𝔤 for all i. Let A_{i} ≡ ∑_{j=0}^{n−1} Q_{ij}k(s_{j}) for each i (Q is the matrix defined in (3.1)). Thus, k(s_{j}) = ∑_{i=0}^{n−1} G(s_{j}, s_{i})A_{i} for j ∈ {0, 1, …, n − 1} . Let l(s) ≡ ∑_{i=0}^{n−1} G(s, s_{i})A_{i} for s ∈ [0, 1]. By the first part of the proof for part (b) we know that l ∈ H_{P}(𝔤). Since, for each s_{j} ∈ P, l(s_{j}) = ∑_{i=0}^{n−1} G(s_{j}, s_{i})A_{i} = k(s_{j}), it follows by part (a) that k − l ≡ 0.

c) Let k ∈ H_{P}(𝔤). Then by part (b), k is of the form k(s) = ∑_{i=0}^{n−1} G(s, s_{i})A_{i}

where A_{i} ∈ 𝔤 for all i. From the definition of G in (2.4) it is clear that G_{ss}(s, s_{i}) = G(s, s_{i}) for each s_{i} ∈ P and s ∉ P. Thus k′′(s) = k(s) for s ∈ [0, 1]\P.

Conversely, suppose that k ∈ H(𝔤) is such that −k′′(s) + k(s) = 0 for all

s ∈ [0, 1]\P. For each h ∈ Ker(π̃_{P}), integration by parts gives

(h, k) = ∫_{0}^{1}{⟨h′(s), k′(s)⟩ + ⟨h(s), k(s)⟩}ds
= ∑_{i=0}^{n−1}{⟨h(s_{i}), k′(s_{i})⟩ − ∫_{s_{i}}^{s_{i+1}}⟨h(s), k′′(s)⟩ds} + ∫_{0}^{1}⟨h(s), k(s)⟩ds
= ∑_{i=0}^{n−1}⟨h(s_{i}), k′(s_{i})⟩ + ∫_{0}^{1}⟨h(s), −k′′(s) + k(s)⟩ds = 0.

Thus, k ∈ (Ker(π̃_{P}))^{⊥} = H_{P}(𝔤). □

Our next goal is to show that π̃_{P} : H_{P}(𝔤) → 𝔤^{P} is an isometric isomorphism.

This will give us a more convenient finite dimensional space, namely H_{P}(𝔤), in which to perform calculations. But first we need to give H_{P}(𝔤) a Lie algebra structure. Note that H_{P}(𝔤) does not inherit the Lie algebra structure from H( 𝔤) since if h, k ∈ H_{P}(𝔤), then [h, k] is not necessarily back in H_{P}(𝔤). We start with the following definitions.

Definition 3.0.2 For each partition P and each h ∈ H(𝔤) (or h ∈ H(R)), define the projection P_{P} : H(𝔤) → H_{P}(𝔤) (or P_{P} : H(R) → H_{P}(R)) by

(P_{P}h)(s) ≡ ∑_{i,j=0}^{n−1} G(s, s_{j})Q_{ij}h(s_{j}).

Note that for each s_{l} ∈ P, (P_{P}h)(s_{l}) = ∑_{i,j=0}^{n−1} G(s_{l}, s_{i})Q_{ij}h(s_{j}) = h(s_{l}) (by

definition of the matrix Q). In other words, P_{P}h and h agree on the partition points of P.

Proposition 3.0.3 For each partition P P_{P} : H(𝔤) → H_{P}(𝔤) is the orthogonal projection onto H_{P}(𝔤). Furthermore, for each h ∈ H(𝔤), P_{P}h is the piece-wise exponential approximation of h that is uniquely determined by its values on P as follows:

(P_{P}h)(t) = \frac{h(s_{i}) + h(s_{i+1})}{ς_{i}} cosh(t− s_{i}) + \frac{Δ_{i}h}{η_{i}} sinh(t− s_{i}) ∀t ∈ [s_{i}, s_{i+1}], (3.2)

where s_{i}≡ \frac{s_{i}+s_{i+1}}{2} , Δ_{i}h ≡ h(s_{i+1}) − h(s_{i}) , ς_{i} ≡ 2 cosh(s_{i}) and η_{i} ≡ 2 sinh(\frac{s_{i+1}−s_{i}}{2} ) .

Proof: Note that for any h ∈ H(𝔤), since (h − P_{P}h)(s_{i}) = h(s_{i}) − (P_{P}h)(s_{i}) =

h(s_{i}) − h(s_{i}) = 0 for all s_{i} ∈ P, then h − P_{P}h ∈ Ker(π̃_{P}) = (H_{P}(𝔤))^{⊥}. This, along with the fact that π̃_{P} is an isometry, shows that H(𝔤) = H_{P}(𝔤)⊕Ker(π̃_{P}) = H_{P}(𝔤)⊕H_{P}(𝔤)^{⊥} since any h ∈ H(𝔤) can be written as h = P_{P}h+(h−P_{P}h) , and if h ∈ H_{P}(𝔤)∩Ker(π̃_{P}), then h(s_{i}) = 0 ∀ s_{i} ∈ P and thus π̃_{P}(h) = 0 ⇒ |h| = |π̃_{P}(h)| = 0 ⇒ h = 0 . Hence, P_{P}h is the orthogonal projection of h onto H_{P}(𝔤).

To see that (3.2) holds, note that P_{P}h is defined in terms of the Green’s function,

G(s, t), which is a sum of exponential functions. Thus, on a subinterval, [s_{i}, s_{i+1}], of the partition, P_{P}h must be exponential of the form (P_{P}h)(t) = A cosh(t− s_{i})+B sinh(t− s_{i}). To determine A and B, we use the fact that (P_{P}h)(s_{i}) = h(s_{i}) and (P_{P}h)(s_{i+1}) = h(s_{i+1}). □

Definition 3.0.4 For h, k ∈ H_{P}(𝔤), define a Lie bracket, [·, ·]_{P}, on H_{P}(𝔤) by [h, k]_{P} ≡ P_{P}[h, k].

We claim that H_{P}(𝔤) is a Lie algebra under [·, ·]_{P}. For this we need to show

the Jacobi identity holds. In other words, for any h, k, l ∈ H_{P}(𝔤), we want

J ≡ [h, [k, l]_{P}]_{P} + [l, [h, k]_{P}]_{P} + [k, [l, h]_{P}]_{P} = 0.

By our remarks in the definition of P_{P}, for each s_{i} ∈ P we have

J(s_{i}) = [h(s_{i}), [k, l](s_{i})] + [l(s_{i}), [h, k](s_{i})] + [k(s_{i}), [l, h](s_{i})]
= 0,

where we have used the Jacobi identity for 𝔤 in the last equality. But since J is in H_{P}(𝔤), by part (a) of Lemma 3.0.1 it is uniquely determined by its values on P and hence J ≡ 0.

The next two propositions verify that π̃_{P} : H_{P}(𝔤) → 𝔤^{P} is indeed an isometric

isomorphism.

Proposition 3.0.5 Given a partition P = {0 = s_{0} < s_{1} < ⋯ < s_{n} = 1}, suppose that h, k ∈ H_{P}(𝔤). Then

(h, k) = ∑_{i,j=1}^{n−1} Q_{ij}⟨h(s_{i}), k(s_{j})⟩,

Thus, the map π̃_{P} : H_{P}(𝔤) → 𝔤^{P} is an isometry, where the inner product on 𝔤^{P} is given in (3.1).

Proof: Suppose k(s) = ∑_{i=0}^{n−1} G(s, s_{i})A_{i} and h(s) = ∑_{i=0}^{n−1} G(s, s_{i})B_{i}, where A_{i}

and B_{i} ∈ 𝔤 for all i. Then

(h, k) = ∑_{i=0}^{n−1}⟨h(s_{i}), A_{i}⟩
= ∑_{i=0}^{n−1}⟨h(s_{i}), ∑_{j=0}^{n−1} Q_{ij}k(s_{j})⟩
= ∑_{i,j=0}^{n−1} Q_{ij}⟨h(s_{i}), k(s_{j})⟩,

where we have used Lemma 2.3.6 for the first equation.□

Corollary 3.0.6 For k(s) = ∑_{i=0}^{n−1} G(s, s_{i})A_{i} and h(s) = ∑_{i=0}^{n−1} G(s, s_{i})B_{i} as above,

(h, k) = ∑_{i,j=0}^{n−1} G(s_{i}, s_{j})⟨B_{i}, A_{j}⟩.

Proof: As in the proof above,

(h, k) = ∑_{j=0}^{n−1}⟨h(s_{j}), A_{j}⟩
= ∑_{j=0}^{n−1}⟨∑_{i=0}^{n−1} G(s_{i}, s_{j})B_{i}, A_{j}⟩
= ∑_{i,j=0}^{n−1} G(s_{i}, s_{j})⟨B_{i}, A_{j}⟩.□

Proposition 3.0.7 The map π̃_{P} : H_{P}(𝔤) → 𝔤^{P} is a Lie algebra isomorphism.

Proof: From the proof of Proposition 3.0.3, it is clear that π̃_{P} is one-to-one.

For any g = (g_{0}, g_{1}, … g_{n−1}) ∈ 𝔤^{P}, choose h ∈ H_{P}(𝔤) to be h(s) ≡ ∑_{i,j=0}^{n−1} G(s, s_{j})Q_{ij}g_{j}.

Then π̃_{P}(h) = g, and thus π̃_{P} is onto. To see that π̃_{P} is a Lie algebra homomorphism, note that for h, k ∈ H_{P}(𝔤),

π̃_{P}([h, k]_{P}) = π̃_{P}(P_{P}[h, k])
= π̃_{P}( ∑_{i,j=0}^{n−1} G(·, s_{j})Q_{ij}[h(s_{j}), k(s_{j})])
= ([h(s_{0}), k(s_{0})], [h(s_{1}), k(s_{1})], …, [h(s_{n−1}), k(s_{n−1})])
= [(h(s_{0}), …, h(s_{n−1})), (k(s_{0}), …, k(s_{n−1}))]_{𝔤^{P}}
= [π̃_{P}(h), π̃_{P}(k)]_{𝔤^{P}}. □

At this point it is useful to discuss the geometry of H_{P}(𝔤).

Definition 3.0.8 Let ∇ be the “Levi-Civita covariant derivative” on H(𝔤). Identifying H_{P}(𝔤) with the left invariant vector fields on G^{P}, we will define on H_{P}(𝔤) a covariant derivative, ∇^{P}, as follows: for each h, k ∈ H_{P}(𝔤) let ∇_{h}^{P}k ≡ P_{P}(∇_{h}k), i.e., ∇_{h}^{P} = P_{P}∇_{h}.

Definition 3.0.9 We define the Laplacian Δ on H(𝔤) by

Δ ≡ ∑_{k∈S} ∇_{k}∇_{k},

and the Laplacian Δ_{P} on H_{P}(𝔤) by

Δ_{P} ≡ ∑_{k∈S_{P}} ∇_{k}^{P}∇_{k}^{P},

where S is a good orthonormal basis for H(𝔤) and S_{P} is any orthonormal basis for H_{P}(𝔤), respectively.

Note that for h ∈ H(𝔤) and l ∈ H_{P}(𝔤),
(P_{P}h, l) = (P_{P}h + h − P_{P}h, l)
= (h, l).

We can use this to show that ∇^{P} is metric compatible based on the metric compatibility of ∇: for all h, k, l ∈ H_{P}(𝔤)

(∇_{h}^{P}k, l) = (P_{P}∇_{h}k, l) = (∇_{h}k, l) = (k, ∇_{h}l) = (k, P_{P}∇_{h}l) = (k, ∇_{h}^{P}l).

Also, since ∇ is torsion free, then so is ∇^{P} since

∇_{h}^{P}k − ∇_{k}^{P}h = P_{P}(∇_{h}k − ∇_{k}h) = P_{P}[h, k] = [h, k]_{P}.

Thus, ∇^{P} deserves to be called the Levi-Civita covariant derivative on H_{P}(𝔤).

For the following lemmas, (·, ·) will denote the inner product on H(R) defined

by (h, h) = ∫_{0}^{1}{(h′(s))^{2} + (h(s))^{2}}ds and S_{P}(R) will be an arbitrary orthonormal basis of H_{P}(R).

Lemma 3.0.10 For all (t_{1}, t_{2}) ∈ [0, 1]^{2}, (P_{P}G(·, t_{1}), G(·, t_{2})) = G_{P}(t_{1}, t_{2}), where

G_{P}(t_{1}, t_{2}) ≡ ∑_{a∈S_{P}(R)} a(t_{1})a(t_{2}).

Proof: Let S_{P}(⊥) be an orthonormal basis of (H_{P}(R))^{⊥} and let S be the

orthonormal basis of H(R) defined by S = S_{P}(R) ∪ S_{P}(⊥). Then

(P_{P}G(·, t_{1}), G(·, t_{2})) = ∑_{a∈S}(P_{P}G(·, t_{1}), a)(G(·, t_{2}), a)
= ∑_{a∈S_{P}(R)}(P_{P}G(·, t_{1}), a)(G(·, t_{2}), a)

+ ∑_{b∈S_{P}(⊥)}(P_{P}G(·, t_{1}), b)(G(·, t_{2}), b)

= ∑_{a∈S_{P}(R)}(G(·, t_{1}), a)(G(·, t_{2}), a)
= ∑_{a∈S_{P}(R)} a(t_{1})a(t_{2}) = G_{P}(t_{1}, t_{2}),

where we have used Lemma 2.3.6 for the last equation.

It is clear from the proof of Lemma 3.0.10 that G_{P}(t_{1}, t_{2}) is well-defined since

the summation in the first equation of the proof is independent of choice of orthonormal basis S.

Lemma 3.0.11 For all (t_{1}, t_{2}) ∈ [0, 1]^{2}, G_{P}(t_{1}, t_{2}) = (P_{P}G(t_{1}, ·))(t_{2}).

Proof: Using Lemma 3.0.10 and the fact that G is a reproducing kernel,

G_{P}(t_{1}, t_{2}) = (P_{P}G(·, t_{1}), G(·, t_{2})) = (P_{P}G(·, t_{1}))(t_{2}) = (P_{P}G(t_{1}, ·))(t_{2}).□

Lemma 3.0.12 For h ∈ L^{1}([0, 1]) and t_{1} ∈ [0, 1],

(P_{P} ∫_{0}^{1} F (·, t_{1})h(t_{1})dt_{1})(t_{2}) = ∫_{0}^{1} −(G_{P})_{s}(t_{1}, t_{2})h(t_{1})dt_{1}

.

Proof:

(P_{P} ∫_{0}^{1} F (·, t_{1})h(t_{1})dt_{1})(t_{2}) = ∑_{i,j=0}^{n−1} G(t_{2}, s_{i})Q_{ij} ∫_{0}^{1} G_{s}(s_{j}, t_{1})h(t_{1})dt_{1}
= ∫_{0}^{1} ∑_{i,j=0}^{n−1} G(t_{2}, s_{i})Q_{ij}G_{s}(s_{j}, t_{1})h(t_{1})dt_{1}
= ∫_{0}^{1} ∑_{i,j=0}^{n−1} −G_{s}(t_{1}, s_{j})Q_{ji}G(s_{i}, t_{2})h(t_{1})dt_{1}
= ∫_{0}^{1} −(G_{P})_{s}(t_{1}, t_{2})h(t_{1})dt_{1}. □

Lemma 3.0.13 Let P = {0 = s_{0} < s_{1} < ⋯ < s_{n−1} < 1} be a given partition as before. Then

G(s, t) = G_{P}(s, t) + ∑_{i=0}^{n−1} G_{i}(s, t),

where

G_{i}(s, t) = \{ \begin{matrix}0 if (s, t) ∉ [s_{i}, s_{i+1}]^{2} \\ \frac{sinh(t−s_{i}) sinh(s_{i+1}−s)}{sinh(s_{i+1}−s_{i})} if s_{i} ≤ t < s < s_{i+1} \\ \frac{sinh(s−s_{i}) sinh(s_{i+1}−t)}{sinh(s_{i+1}−s_{i})} if s_{i} ≤ s ≤ t < s_{i+1} \end{matrix}

Proof: Let t ∈ [0, 1] be fixed. Since G(·, t) ∈ H(R) = H_{P}(R) ⊕ Ker(π̃_{P}), and

since, by Lemma 3.0.11, (P_{P}G(·, t))(s) = G_{P}(s, t), then we can write

G(s, t) = G_{P}(s, t) + V (s, t) ∀ s ∈ [0, 1]

where V (·, t) ∈ Ker(π̃_{P}) (i.e., V (s_{i}, t) = 0 ∀ s_{i} ∈ P). Assume t ∈ [s_{i}, s_{i+1}] for some i. Define

G_{i}(s, t) ≡ G(s, t) − G_{P}(s, t) ∀ s ∈ [s_{i}, s_{i+1}].

Since G_{P}(s, t) = (P_{P}G(·, t))(s) = ∑_{j,l=0}^{n−1} G(s, s_{j})Q_{jl}G(s_{l}, t), then G_{P}(s_{i}, t) = G(s_{i}, t) and

G_{P}(s_{i+1}, t) = G(s_{i+1}, t), so G_{i}(s_{i}, t) = G_{i}(s_{i+1}, t) = 0.

Let L be the differential operator defined by L ≡ −\frac{d^{2}}{ds^{2}} + 1. Since we know that

L(G(·, s_{j}))(s) = 0 for s ≠ s_{j}, then L(G_{i}(·, t))(s) = 0 for s_{i} < s < t and t < s < s_{i+1}. Since G_{s}(t^{+}, t) − G_{s}(t^{−}, t) = −1 and G_{s}(s, s_{j}) is continuous on (s_{i}, s_{i+1}) for all j, then (G_{i})_{s}(t^{+}, t) − (G_{i})_{s}(t^{−}, t) = −1. Also, since G is continuous, then G_{i} is continuous.

Putting all of the above facts together, we see that G_{i} must be the Green’s

function on [s_{i}, s_{i+1}] satisfying the following conditions:

1. L(G_{i}(·, t))(s) = 0 for s ∈ (s_{i}, t) ∪ (t, s_{i+1})

2. G_{i}(s_{i}, t) = G_{i}(s_{i+1}, t) = 0

3. (G_{i})_{s}(t^{+}, t) − (G_{i})_{s}(t^{−}, t) = −1

4. G_{i}(·, t) is continuous on [s_{i}, s_{i+1}].

Using the Green’s function method of solving linear differential equations with

linear boundary conditions (see p. 225 in Zwillinger [21]]), we use (1) above to determine that G_{i} must be of the form

G_{i}(s, t) = \{ \begin{matrix}Ae^{s} + Be^{−s} for s_{i} ≤ t < s \\ Ee^{s} + F e^{−s} for s ≤ t ≤ s_{i+1}. \end{matrix}

From (1)-(4) above, we get the following system of equations:

Ee^{s_{i}} + F e^{−s_{i}} = 0
Ae^{s_{i+1}} + Be^{−s_{i+1}} = 0
Ae^{t} − Be^{−t} − Ee^{t} + F e^{−t} = −1
Ae^{t} + Be^{−t} = Ee^{t} + F e^{−t}.

Solving this system of linear equations for A, B, E and F will give

A = −\frac{1}{2}e^{−t} + \frac{e^{−s_{i}} sinh(s_{i+1} − t)}{2 sinh(s_{i+1} − s_{i})}
B = \frac{1}{2}e^{t} − \frac{e^{s_{i}} sinh(s_{i+1} − t)}{2 sinh(s_{i+1} − s_{i})}
E = \frac{e^{−s_{i}} sinh(s_{i+1} − t)}{2 sinh(s_{i+1} − s_{i})}
F = −\frac{e^{−s_{i}} sinh(s_{i+1} − t)}{2 sinh(s_{i+1} − s_{i})}.

Substituting these terms in our formula for G_{i}(s, t) then gives

G_{i}(s, t) = \{ \begin{matrix}\{ \begin{matrix}−\frac{1}{2}e^{s−t} + \frac{e^{s−s_{i}} sinh(s_{i+1}−t)}{2 sinh(s_{i+1}−s_{i})} \\ +\frac{1}{2}e^{−s+t} − \frac{e^{−s+s_{i}} sinh(s_{i+1}−t)}{2 sinh(s_{i+1}−s_{i})} \end{matrix} \} for s_{i} ≤ t < s ≤ s_{i+1} \\ \frac{e^{s−s_{i}} sinh(s_{i+1}−t)}{2 sinh(s_{i+1}−s_{i})} − \frac{e^{−s+s_{i}} sinh(s_{i+1}−t)}{2 sinh(s_{i+1}−s_{i})} for s_{i} ≤ s ≤ t ≤ s_{i+1} \end{matrix}

= \{ \begin{matrix}sinh(t − s) + \frac{sinh(s−s_{i})}{sinh(s_{i+1}−s_{i})} sinh(s_{i+1} − t) for s_{i} ≤ t < s ≤ s_{i+1} \\ \frac{sinh(s−s_{i})}{sinh(s_{i+1}−s_{i})} sinh(s_{i+1} − t) for s_{i} ≤ s ≤ t ≤ s_{i+1} \end{matrix} .

For the part of G_{i}(s, t) where s_{i} ≤ t < s ≤ s_{i+1}, we use the identity sinh(a) sinh(b) = \frac{1}{2}(cosh(a + b) − cosh(a − b)) twice as follows:

sinh(t − s) + \frac{sinh(s − s_{i})}{sinh(s_{i+1} − s_{i})} sinh(s_{i+1} − t)

= \frac{sinh(t − s) sinh(s_{i+1} − s_{i}) + sinh(s − s_{i}) sinh(s_{i+1} − t)}{sinh(s_{i+1} − s_{i})}

= \frac{\frac{1}{2} \{ \begin{matrix}cosh(t − s + s_{i+1} − s_{i}) − cosh(t − s − s_{i+1} + s_{i}) \\ + cosh(s − s_{i} + s_{i+1} − t) − cosh(s − s_{i} − s_{i+1} + t) \end{matrix} \}}{sinh(s_{i+1} − s_{i})}
= \frac{\frac{1}{2}\{cosh(t − s_{i} + s_{i+1} − s) − cosh(t − s_{i} − (s_{i+1} − s))\}}{sinh(s_{i+1} − s_{i})}
= \frac{sinh(t − s_{i}) sinh(s_{i+1} − s)}{sinh(s_{i+1} − s_{i})}.

Hence, G_{i}(s, t) may now be written as

G_{i}(s, t) = \{ \begin{matrix}\frac{sinh(t−s_{i}) sinh(s_{i+1}−s)}{sinh(s_{i+1}−s_{i})} for s_{i} ≤ t < s ≤ s_{i+1} \\ \frac{sinh(s−s_{i}) sinh(s_{i+1}−t)}{sinh(s_{i+1}−s_{i})} for s_{i} ≤ s ≤ t ≤ s_{i+1} \end{matrix} .

The following verifies that conditions (1)-(4) are satisfied by G_{i}.

1. Since L(sinh(s_{i+1}−·))(s) = − sinh(s_{i+1}−s)+sinh(s_{i+1}−s) = 0 and L(sinh(·−s_{i}))(s) = − sinh(s − s_{i}) + sinh(s − s_{i}) = 0, then clearly L(G_{i}(·, t))(s) = 0 for s ∈ (s_{i}, t) ∪ (t, s_{i+1}). 2.

G_{i}(s_{i}, t) = \frac{sinh(s_{i} − s_{i}) sinh(s_{i+1} − t)}{sinh(s_{i+1} − s_{i})} = 0
G_{i}(s_{i+1}, t) = \frac{sinh(t − s_{i}) sinh(s_{i+1} − s_{i+1})}{sinh(s_{i+1} − s_{i})} = 0

3.

(G_{i})_{s}(t^{+}, t) = −\frac{sinh(t − s_{i}) cosh(s_{i+1} − t)}{sinh(s_{i+1} − s_{i})}
= − \frac{\frac{1}{2}\{sinh(s_{i+1} − s_{i}) + sinh(2t − (s_{i+1} + s_{i}))\}}{sinh(s_{i+1} − s_{i})}
(G_{i})_{s}(t^{−}, t) = \frac{cosh(t − s_{i}) sinh(s_{i+1} − t)}{sinh(s_{i+1} − s_{i})}
= \frac{\frac{1}{2}\{sinh(s_{i+1} − s_{i}) + sinh(−2t + (s_{i+1} + s_{i}))\}}{sinh(s_{i+1} − s_{i})}.

Thus, (G_{i})_{s}(t^{+}, t) − (G_{i})_{s}(t^{−}, t) = −\frac{1}{2} − \frac{1}{2} = −1 .

4. Since sinh is continuous, we just need to check the continuity of G_{i}(s, t) at s = t. Indeed,

G_{i}(t^{−}, t) = \frac{sinh(t − s_{i}) sinh(s_{i+1} − t)}{sinh(s_{i+1} − s_{i})} = G_{i}(t^{+}, t).

We have thus shown that for (s, t) ∈ [s_{i}, s_{i+1}], V (s, t) = G(s, t) − G_{P}(s, t) =

G_{i}(s, t), where G_{i}(s, t) is defined as above. If t ∈ [s_{i}, s_{i+1}] for some i, then on each interval [s_{j}, s_{j+1}] where i ≠ j, it is easy to see that V (·, t) must be of the form V (s, t) = Ae^{s}+Be^{−s} with V (s_{j}, t) = V (s_{j+1}, t) = 0. Thus A = B = 0, and so V (s, t) = 0 for s ∈ [s_{j}, s_{j+1}]. We can then conclude that

V (s, t) = \{ \begin{matrix}0 for s ∉ [s_{i}, s_{i+1}] \\ G_{i}(s, t) for s ∈ [s_{i}, s_{i+1}] \end{matrix} .

Extending G_{i}(s, t) to be zero outside of [s_{i}, s_{i+1}], we can then write V (s, t) = ∑_{i=0}^{n−1} G_{i}(s, t)

∀ s, t ∈ [0, 1] and thus G(s, t) = G_{P}(s, t) + ∑_{i=0}^{n−1} G_{i}(s, t). □

One clearly sees by the formula for each G_{i}(s, t) that G_{i} is symmetric since G

and G_{P} are both symmetric. Using our convention that (G_{i})_{s} is the partial derivative of G_{i} with respect to its first component and (G_{i})_{t} is the partial derivative of G_{i} with respect to its second component we have the following for (x, y) ∈ [s_{i}, s_{i+1}]^{2}:

(G_{i})_{s}(x, y) = \{ \begin{matrix}−\frac{sinh(y−s_{i}) cosh(s_{i+1}−x)}{sinh(s_{i+1}−s_{i})} for s_{i} ≤ y < x ≤ s_{i+1} \\ \frac{cosh(x−s_{i}) sinh(s_{i+1}−y)}{sinh(s_{i+1}−s_{i})} for s_{i} ≤ x ≤ y ≤ s_{i+1} \end{matrix}
= \{ \begin{matrix}\frac{sinh(s_{i}−y) cosh(s_{i+1}−x)}{sinh(s_{i+1}−s_{i})} for s_{i} ≤ y < x ≤ s_{i+1} \\ \frac{cosh(x−s_{i}) sinh(s_{i+1}−y)}{sinh(s_{i+1}−s_{i})} for s_{i} ≤ x ≤ y ≤ s_{i+1} \end{matrix} .

Using the identity sinh(a + b) = sinh(a) cosh(b) + cosh(a) sinh(b), we then get

(G_{i})_{s}(x, y) = \{ \begin{matrix}\frac{sinh(−(y+x)+s_{i+1}+s_{i})−cosh(s_{i}−y) sinh(s_{i+1}−x)}{sinh(s_{i+1}−s_{i})} for s_{i} ≤ y < x ≤ s_{i+1} \\ \frac{sinh(−(y+x)+s_{i+1}+s_{i})−sinh(s_{i}−x) cosh(s_{i+1}−y)}{sinh(s_{i+1}−s_{i})} for s_{i} ≤ x ≤ y ≤ s_{i+1} \end{matrix} .

Noting that

(G_{i})_{t}(x, y) = \{ \begin{matrix}\frac{cosh(y−s_{i}) sinh(s_{i+1}−x)}{sinh(s_{i+1}−s_{i})} for s_{i} ≤ y < x ≤ s_{i+1} \\ \frac{sinh(s_{i}−x) cosh(s_{i+1}−y)}{sinh(s_{i+1}−s_{i})} for s_{i} ≤ x ≤ y ≤ s_{i+1} \end{matrix}

we then have the following corollary. Corollary 3.0.14

(G_{i})_{s}(x, y) = \{ \begin{matrix}−(G_{i})_{t}(x, y) + \frac{sinh(−(y+x)+s_{i+1}+s_{i})}{sinh(s_{i+1}−s_{i})} for (x, y) ∈ [s_{i}, s_{i+1}]^{2} \\ 0 for (x, y) ∉ [s_{i}, s_{i+1}]^{2} \end{matrix}

= \{ \begin{matrix}−(G_{i})_{s}(y, x) + \frac{sinh(−(y+x)+s_{i+1}+s_{i})}{sinh(s_{i+1}−s_{i})} for (x, y) ∈ [s_{i}, s_{i+1}]^{2} \\ 0 for (x, y) ∉ [s_{i}, s_{i+1}]^{2} \end{matrix} .

3.1 Convergence of Finite Dimensional Laplacian

Theorem 3.1.1

∥P_{P}( Δ − Δ_{P})P_{P}∥_{op} → 0 as |P| → 0.

Proof: Our goal is to find, for each partition P, an ϵ(P) such that ϵ(P) → 0 as

|P| → 0 and, for each h, J ∈ H_{P}(𝔤) ,

|(( Δ − Δ_{P})h, J)| ≤ ϵ(P) ∥h∥ ∥J∥ .  (3.3)

Due to the length of the calculations that will unfold for the proof of this

theorem, we will use several lemmas to focus on separate estimations which combined will give the inequality in (3.3). Since we are really comparing (Δh, J) with (Δ_{P}h, J), we start by expanding these two terms in the following two lemmas. Lemma 3.1.2 For h, J ∈ H(𝔤) ,

(Δh, J) = − ∫_{0}^{1} a cosh(\frac{1}{2})K⟨h′(t), J′(t)⟩dt  (3.4)

+\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} G(x, t)^{2}K⟨h′(x), J′(t)⟩dxdt

−\frac{1}{4} ∫_{0}^{1} ∫_{0}^{1} G(x, t)^{2}K⟨h(x), J(t)⟩dxdt.

Proof: Letting S be an orthonormal basis of H(𝔤) of the form S = h_{0}𝔤_{0} where

h_{0} is an orthonormal basis of H(R) and 𝔤_{0} is an orthonormal basis of 𝔤, we have:

(Δh, J)

= ∑_{k∈S} −(∇_{k}h, ∇_{k}J)
= − ∑_{k∈S} ∫_{0}^{1}{⟨(∇_{k}h)′(x), (∇_{k}J)′(x)⟩ + ⟨(∇_{k}h)(x), (∇_{k}J)(x)⟩}dx
= − ∑_{k∈S} ∫_{0}^{1}\{⟨[k, h′](x) − ∫_{0}^{1} G(x, t)[k, h′]dt + \frac{1}{2} ∫_{0}^{1} F (x, t)[k, h]dt,

[k, J′](x) − ∫_{0}^{1} G(x, t)[k, J′]dt + \frac{1}{2} ∫_{0}^{1} F (x, t)[k, J]dt⟩

+⟨− ∫_{0}^{1} F (x, t)[k, h′]dt + \frac{1}{2} ∫_{0}^{1} G(x, t)[k, h]dt,

− ∫_{0}^{1} F (x, t)[k, J′]dt + \frac{1}{2} ∫_{0}^{1} G(x, t)[k, J]dt⟩\}dx

= − ∑_{a∈h_{0}} ∑_{B∈𝔤_{0}}{∫_{0}^{1}⟨[a(x)B, h′(x)], [a(x)B, J′(x)]⟩dx

+ ∫_{0}^{1}⟨− ∫_{0}^{1} G(x, t)[a(t)B, h′(t)]dt + \frac{1}{2} ∫_{0}^{1} F (x, t)[a(t)B, h(t)]dt, [a(x)B, J′(x)]⟩dx

+ ∫_{0}^{1}⟨[a(x)B, h′(x)], − ∫_{0}^{1} G(x, t)[a(t)B, J′(t)]dt + \frac{1}{2} ∫_{0}^{1} F (x, t)[a(t)B, J(t)]dt⟩dx

+ ∫_{0}^{1}⟨− ∫_{0}^{1} G(x, t)[a(t)B, h′(t)]dt + \frac{1}{2} ∫_{0}^{1} F (x, t)[a(t)B, h(t)]dt,

− ∫_{0}^{1} G(x, t)[a(t)B, J′(t)]dt + \frac{1}{2} ∫_{0}^{1} F (x, t)[a(t)B, J(t)]dt⟩dx

+ ∫_{0}^{1}⟨− ∫_{0}^{1} F (x, t)[a(t)B, h′(t)]dt + \frac{1}{2} ∫_{0}^{1} G(x, t)[a(t)B, h(t)]dt,

− ∫_{0}^{1} F (x, t)[a(t)B, J′(t)]dt + \frac{1}{2} ∫_{0}^{1} G(x, t)[a(t)B, J(t)]dt⟩dx\}

= − ∫_{0}^{1}( ∑_{a∈h_{0}} a(x)(x)) ∑_{B∈𝔤_{0}}⟨[B, h′(x)], [B, J′(x)]⟩dx

− ∫_{0}^{1} ∫_{0}^{1}( ∑_{a∈h_{0}} a(t)(x)){−G(x, t) ∑_{B∈𝔤_{0}}⟨[B, h′(t)], [B, J′(x)]⟩

+\frac{1}{2}F (x, t) ∑_{B∈𝔤_{0}}⟨[B, h(t)], [B, J′(x)]⟩\}dtdx

− ∫_{0}^{1} ∫_{0}^{1}( ∑_{a∈h_{0}} a(x)(t)){−G(x, t) ∑_{B∈𝔤_{0}}⟨[B, h′(t)], [B, J′(x)]⟩

+\frac{1}{2}F (x, t) ∑_{B∈𝔤_{0}}⟨[B, h′(t)], [B, J(x)]⟩\}dtdx

− ∫_{0}^{1} ∫_{0}^{1} ∫_{0}^{1}( ∑_{a∈h_{0}} a(t_{1})(t_{2})){G(x, t_{1})G(x, t_{2}) ∑_{B∈𝔤_{0}}⟨[B, h′(t_{1})], [B, J′(t_{2})]⟩

−\frac{1}{2}G(x, t_{1})F (x, t_{2}) ∑_{B∈𝔤_{0}}⟨[B, h′(t_{1})], [B, J(t_{2})]⟩

−\frac{1}{2}F (x, t_{1})G(x, t_{2}) ∑_{B∈𝔤_{0}}⟨[B, h(t_{1})], [B, J′(t_{2})]⟩

+\frac{1}{4}F (x, t_{1})F (x, t_{2}) ∑_{B∈𝔤_{0}}⟨[B, h(t_{1})], [B, J(t_{2})]⟩

+F (x, t_{1})F (x, t_{2}) ∑_{B∈𝔤_{0}}⟨[B, h′(t_{1})], [B, J′(t_{2})]⟩

−\frac{1}{2}G(x, t_{1})F (x, t_{2}) ∑_{B∈𝔤_{0}}⟨[B, h(t_{1})], [B, J′(t_{2})]⟩

−\frac{1}{2}F (x, t_{1})G(x, t_{2}) ∑_{B∈𝔤_{0}}⟨[B, h′(t_{1})], [B, J(t_{2})]⟩

+\frac{1}{4}G(x, t_{1})G(x, t_{2}) ∑_{B∈𝔤_{0}}⟨[B, h(t_{1})], [B, J(t_{2})]⟩\}dt_{1}dt_{2}dx

= − ∫_{0}^{1}(G(x, x)K⟨h′(x), J′(x)⟩dx

+ ∫_{0}^{1} ∫_{0}^{1} G(t, x)\{G(x, t)K⟨h′(t), J′(x)⟩ − \frac{1}{2}F (x, t)K⟨h(t), J′(x)⟩\}dtdx

+ ∫_{0}^{1} ∫_{0}^{1} G(t, x)\{G(x, t)K⟨h′(t), J′(x)⟩ − \frac{1}{2}F (x, t)K⟨h′(t), J(x)⟩\}dtdx

− ∫_{0}^{1} ∫_{0}^{1} ∫_{0}^{1} G(t_{1}, t_{2}){G(x, t_{1})G(x, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩

−\frac{1}{2}G(x, t_{1})F (x, t_{2})K⟨h′(t_{1}), J(t_{2})⟩

−\frac{1}{2}F (x, t_{1})G(x, t_{2})K⟨h(t_{1}), J′(t_{2})⟩ + \frac{1}{4}F (x, t_{1})F (x, t_{2})K⟨h(t_{1}), J(t_{2})⟩

+F (x, t_{1})F (x, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩ − \frac{1}{2}G(x, t_{1})F (x, t_{2})K⟨h′(t_{1}), J(t_{2})⟩

−\frac{1}{2}F (x, t_{1})G(x, t_{2})K⟨h(t_{1}), J′(t_{2})⟩ + \frac{1}{4}G(x, t_{1})G(x, t_{2})K⟨h(t_{1}), J(t_{2})⟩\}dt_{1}dt_{2}dx

= − ∫_{0}^{1}(G(x, x)K⟨h′(x), J′(x)⟩dx + 2 ∫_{0}^{1} ∫_{0}^{1} G(x, t)^{2}K⟨h′(x), J′(t)⟩dtdx

+\frac{1}{4} ∫_{0}^{1} ∫_{0}^{1} \frac{d}{dt}(G(x, t))^{2}K⟨h(x), J′(t)⟩dtdx

+\frac{1}{4} ∫_{0}^{1} ∫_{0}^{1} \frac{d}{dt}(G(x, t))^{2}K⟨h′(x), J(t)⟩dtdx

− ∫_{0}^{1} ∫_{0}^{1} G(t_{1}, t_{2}){∫_{0}^{1} G(x, t_{1})G(x, t_{2})dx · K⟨h′(t_{1}), J′(t_{2})⟩

−\frac{1}{4} ∫_{0}^{1} G_{xx}(x, t_{1})G(x, t_{2})dx · K⟨h(t_{1}), J(t_{2})⟩

− ∫_{0}^{1} G_{xx}(x, t_{1})G(x, t_{2})dx · K⟨h′(t_{1}), J′(t_{2})⟩

−\frac{1}{4} ∫_{0}^{1} G(x, t_{1})G(x, t_{2})dx · K⟨h(t_{1}), J(t_{2})⟩\}dt_{1}dt_{2}

= − ∫_{0}^{1}(G(x, x)K⟨h′(x), J′(x)⟩dx + 2 ∫_{0}^{1} ∫_{0}^{1} G(x, t)^{2}K⟨h′(x), J′(t)⟩dtdx

−\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} G(x, t)^{2}K⟨h′(t), J′(x)⟩dtdx

− ∫_{0}^{1} ∫_{0}^{1} G(t_{1}, t_{2})\{G(t_{1}, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩ + \frac{1}{4}G(t_{1}, t_{2})K⟨h(t_{1}), J(t_{2})⟩\}dt_{1}dt_{2}.

Renaming variables and noting that G(x, x) = a cosh(\frac{1}{2}), we see that (3.4) holds.□ Lemma 3.1.3 For a given partition P = {0 = s_{0} < s_{1} < ⋯ < s_{n−1} < 1} and h, J ∈ H_{P}(𝔤),

(Δ_{P}h, J)

= − ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} cosh(s_{i} − t_{1} + s_{i+1} − t_{2})G_{P}(t_{1}, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}

+\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} G_{P}^{2}(t_{1}, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}

−\frac{1}{4} ∫_{0}^{1} ∫_{0}^{1} G_{P}^{2}(t_{1}, t_{2})K⟨h(t_{1}), J(t_{2})⟩dt_{1}dt_{2}

− ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} sinh(−(t_{2} + t_{1}) + s_{i+1} + s_{i})

.G_{P}(t_{1}, t_{2})(K⟨h′(t_{1}), J(t_{2})⟩ + K⟨h(t_{1}), J′(t_{2})⟩)dt_{1}dt_{2}.  (3.5)

Proof: Let S_{P} be an orthonormal basis of H_{P}(𝔤) of the form S = h_{P} 𝔤_{0} where h_{P} is an orthonormal basis of H_{P}(R) and 𝔤_{0} is an orthonormal basis of 𝔤, Then

(Δ_{P}h, J) = − ∑_{k∈S_{P}}(∇_{k}^{P}h, ∇_{k}^{P}J)  (3.6)
= − ∑_{k∈S_{P}}(−P_{P}(∫_{0}^{1} F (·, t_{1})[k, h′](t_{1})dt_{1}) + \frac{1}{2}P_{P}(∫_{0}^{1} G(·, t_{1})[k, h](t_{1})dt_{1}),

−P_{P}(∫_{0}^{1} F (·, t_{2})[k, J′](t_{2})dt_{2}) + \frac{1}{2}P_{P}(∫_{0}^{1} G(·, t_{2})[k, J](t_{2})dt_{2}))

= − ∑_{k∈S_{P}}{(∫_{0}^{1}(G_{P})_{s}(t_{1}, ·)[k, h′](t_{1})dt_{1}, ∫_{0}^{1}(G_{P})_{s}(t_{2}, ·)[k, J′](t_{2})dt_{2})

+\frac{1}{2}(∫_{0}^{1}(G_{P})_{s}(t_{1}, ·)[k, h′](t_{1})dt_{1}, ∫_{0}^{1} G_{P}(·, t_{2})[k, J](t_{2})dt_{2})

+\frac{1}{2}(∫_{0}^{1} G_{P}(·, t_{1})[k, h](t_{1})dt_{1}, ∫_{0}^{1}(G_{P})_{s}(t_{2}, ·)[k, J′](t_{2})dt_{2})

+\frac{1}{4}(∫_{0}^{1} G_{P}(·, t_{1})[k, h](t_{1})dt_{1}, ∫_{0}^{1} G_{P}(·, t_{2})[k, J](t_{2})dt_{2})\},  (3.7)

where we have used Lemmas (3.0.11) and (3.0.12) for the last equation. We will write the last summation in (3.7) as

− ∑_{k∈S_{P}}\{A_{k} + \frac{1}{2}B_{k} + \frac{1}{2}C_{k} + \frac{1}{4}D_{k}\}  (3.8)

and calculate each of the inner products, A_{k}, B_{k}, C_{k}, and D_{k}, individually in what follows.

First, using Lemma 3.0.10, we have

∑_{k∈S_{P}} A_{k}

= ∑_{k∈S_{P}}(∫_{0}^{1}( ∑_{l∈h_{P}} l′(t_{1})l(·))[k, h′](t_{1})dt_{1}, ∫_{0}^{1}( ∑_{l̃∈h_{P}} l̃′(t_{2})l̃(·))[k, J′](t_{2})dt_{2}  (3.9)
= ∑_{k∈S_{P}}\{∫_{0}^{1}⟨\frac{d}{dx} ∫_{0}^{1}( ∑_{l∈h_{P}} l′(t_{1})l(x))[k, h′](t_{1})dt_{1}, \frac{d}{dx} ∫_{0}^{1}( ∑_{l̃∈h_{P}}^{′} l̃′(t_{2})l̃(x))[k, J′](t_{2})dt_{2}⟩dx

+ ∫_{0}^{1}⟨∫_{0}^{1}( ∑_{l∈h_{P}} l′(t_{1})l(x))[k, h′](t_{1})dt_{1}, ∫_{0}^{1}( ∑_{l̃∈h_{P}}^{′} l̃′(t_{2})l̃(x))[k, J′](t_{2})dt_{2}⟩dx}

= ∑_{k∈S_{P}} ∑_{l,l̃∈h_{P}}{∫_{0}^{1}⟨∫_{0}^{1} l′(t_{1})l′(x)[k, h′](t_{1})dt_{1}, ∫_{0}^{1} l̃′(t_{2})l̃(x)[k, J′](t_{2})dt_{2}⟩dx

+ ∫_{0}^{1}⟨∫_{0}^{1} l′(t_{1})l(x)[k, h′](t_{1})dt_{1}, ∫_{0}^{1} l̃′(t_{2})l̃(x)[k, J′](t_{2})dt_{2}⟩dx}

= ∑_{k∈S_{P}} ∑_{l,l̃∈h_{P}} ∫_{0}^{1}{l′(x)l̃′(x) + l(x)l̃(x)}dx · ⟨∫_{0}^{1} l′(t_{1})[k, h′](t_{1})dt_{1}, ∫_{0}^{1} l̃′(t_{2})[k, J′[(t_{2}]dt_{2}⟩
= ∑_{k∈S_{P}} ∑_{l,l̃∈h_{P}}(l(·), l̃(·)) · ⟨∫_{0}^{1} l′(t_{1})[k, h′](t_{1})dt_{1}, ∫_{0}^{1} l̃′(t_{2})[k, J′[(t_{2}]dt_{2}⟩
= ∫_{0}^{1} ∫_{0}^{1}( ∑_{l∈h_{P}} l′(t_{1})l′(t_{2})) ∑_{a∈h_{P}} ∑_{B∈𝔤_{0}}⟨[a(t_{1})B, h′(t_{1})], [a(t_{2})B, J′(t_{2})]⟩dt_{1}dt_{2}
= ∫_{0}^{1} ∫_{0}^{1}(G_{P})_{st}(t_{1}, t_{2})G_{P}(t_{1}, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}  (3.10)

where we have used the fact that h_{P} is an orthonormal basis of H_{P}(R) to get the second to last equation above.

At this point we make note that (G_{P})_{st}(t_{1}, t_{2}) can be expressed as follows:

(G_{P})_{st}(t_{1}, t_{2})

= \{ \begin{matrix}\{ \begin{matrix}−a cosh(t_{1} − t_{2} + \frac{1}{2}) \\ + ∑_{i=0}^{n−1} a_{i} cosh(s_{i} − t_{1}) cosh(s_{i+1} − t_{2})1_{[s_{i},s_{i+1}]^{2}}(t_{1}, t_{2}) \end{matrix} \} if t_{1} ≤ t_{2} \\ \{ \begin{matrix}−a cosh(t_{1} − t_{2} − \frac{1}{2}) \\ + ∑_{i=0}^{n−1} a_{i} cosh(s_{i} − t_{2}) cosh(s_{i+1} − t_{1})1_{[s_{i},s_{i+1}]^{2}}(t_{1}, t_{2}) \end{matrix} \} if t_{2} < t_{1} \end{matrix}

= \{ \begin{matrix}\{ \begin{matrix}−a cosh(t_{1} − t_{2} + \frac{1}{2}) + ∑_{i=0}^{n−1}\{−a_{i} sinh(s_{i} − t_{1}) sinh(s_{i+1} − t_{2}) \\ +a_{i} cosh(s_{i} − t_{1} + s_{i+1} − t_{2})\}1_{[s_{i},s_{i+1}]^{2}} (t_{1}, t_{2}) \end{matrix} \} if t_{1} ≤ t_{2} \\ \{ \begin{matrix}−a cosh(t_{1} − t_{2} − \frac{1}{2}) + ∑_{i=0}^{n−1}\{−a_{i} sinh(s_{i} − t_{2}) sinh(s_{i+1} − t_{1}) \\ +a_{i} cosh(s_{i} − t_{2} + s_{i+1} − t_{1})\}1_{[s_{i},s_{i+1}]^{2}} (t_{1}, t_{2}) \end{matrix} \} if t_{2} < t_{1} \end{matrix}

= −G_{P}(t_{1}, t_{2}) + ∑_{i=0}^{n−1} a_{i} cosh(s_{I} − t_{1} + s_{i+1} − t_{2})1_{[s_{i},s_{i+1}]^{2}} (t_{1}, t_{2})

We can then conclude that

∑_{k∈S_{P}} A_{k}

= ∫_{0}^{1} ∫_{0}^{1} −G_{P}^{2}(t_{1}, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}

+ ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} cosh(s_{I} − t_{1} + s_{i+1} − t_{2})G_{P}(t_{1}, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}.

For the second term in (3.8), the same reasoning used in (3.10) gives

∑_{k∈S_{P}} B_{k} = ∫_{0}^{1} ∫_{0}^{1}(G_{P})_{s}(t_{1}, t_{2})G_{P}(t_{1}, t_{2})K⟨h′(t_{1}), J(t_{2})⟩dt_{1}dt_{2}.

But by Corollary 3.0.14,

(G_{P})_{s}(t_{1}, t_{2}) = G_{s}(t_{1}, t_{2}) − ∑_{i=0}^{n−1}(G_{i})_{s}(t_{1}, t_{2})
= −G_{t}(t_{1}, t_{2})

+ ∑_{i=0}^{n−1}{(G_{i})_{t}(t_{1}, t_{2}) − a_{i} sinh(−(t_{1} + t_{2}) + s_{i+1} + s_{i})}1_{[s_{i},s_{i+1}]^{2}}(t_{1}, t_{2}).

Thus

∑_{k∈S_{P}} B_{k}

= ∫_{0}^{1} ∫_{0}^{1} −(G_{P})_{t}(t_{1}, t_{2})G_{P}(t_{1}, t_{2})K⟨h′(t_{1}), J(t_{2})⟩dt_{1}dt_{2}

− ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} sinh(−(t_{1} + t_{2}) + s_{i+1} + s_{i})G_{P}(t_{1}, t_{2})K⟨h′(t_{1}), J(t_{2})⟩dt_{1}dt_{2}

= −\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} \frac{d}{dt_{2}} (G_{P}(t_{1}, t_{2}))^{2}K⟨h′(t_{1}), J(t_{2})⟩dt_{1}dt_{2}

− ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} sinh(−(t_{1} + t_{2}) + s_{i+1} + s_{i})G_{P}(t_{1}, t_{2})K⟨h′(t_{1}), J(t_{2})⟩dt_{1}dt_{2}

= \frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} G_{P}^{2}(t_{1}, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}

− ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} sinh(−(t_{1} + t_{2}) + s_{i+1} + s_{i})G_{P}(t_{1}, t_{2})K⟨h′(t_{1}), J(t_{2})⟩dt_{1}dt_{2},

where we have used integration by parts for the last equation.

For the third term in (3.8), the same reasoning used above for the second term

gives

∑_{k∈S_{P}} C_{k}

= \frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} G_{P}^{2}(t_{1}, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}

− ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} sinh(−(t_{1} + t_{2}) + s_{i+1} + s_{i})G_{P}(t_{1}, t_{2})K⟨h(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}

Finally, the forth term in (3.8) can be calculated as follows:

∑_{k∈S_{P}} D_{k}

= ∑_{k∈S_{P}}{∫_{0}^{1}⟨∫_{0}^{1}(G _{P})_{s}(x, t_{1})[k, h](t_{1})dt_{1}, ∫_{0}^{1}(G_{P})_{s}(x, t_{2})[k, J](t_{2})dt_{2}⟩dx

+ ∫_{0}^{1}⟨∫_{0}^{1} G_{P}(x, t_{1})[k, h](t_{1})dt_{1}, ∫_{0}^{1} G_{P}(x, t_{2})[k, J](t_{2})dt_{2}⟩dx

= ∫_{0}^{1}{∫_{0}^{1}(G_{P})_{s}(x, t_{1})(G_{P})_{s}(x, t_{2})dx

+ ∫_{0}^{1} G_{P}(x, t_{1})G_{P}(x, t_{2})dx}G_{P}(t_{1}, t_{2})K⟨h(t_{1}), J(t_{2})⟩dt_{1}dt_{2}

= ∫_{0}^{1} ∫_{0}^{1}(G_{P}(·, t_{1}), G_{P}(·, t_{2}))G_{P}(t_{1}, t_{2})K⟨h(t_{1}), J(t_{2})⟩dt_{1}dt_{2}
= ∫_{0}^{1} ∫_{0}^{1} G_{P}(t_{1}, t_{2})^{2}K⟨h(t_{1}), J(t_{2})⟩dt_{1}dt_{2},

where we have used Lemmas (3.0.10) and (3.0.11) for the last equation.

Having calculated the four terms in (3.8), we can now conclude that (Δ_{P}h, J)

= − ∑_{k∈S_{P}}\{A_{k} + \frac{1}{2}B_{k} + \frac{1}{2}C_{k} + \frac{1}{4}D_{k}\}
= ∫_{0}^{1} ∫_{0}^{1} G_{P}^{2}(t_{1}, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}

− ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} cosh(s_{I} − t_{1} + s_{i+1} − t_{2})G_{P}(t_{1}, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}

−\frac{1}{4} ∫_{0}^{1} ∫_{0}^{1} G_{P}^{2}(t_{1}, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}

−\frac{1}{2} ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} sinh(−(t_{1} + t_{2}) + s_{i+1} + s_{i})G_{P}(t_{1}, t_{2})K⟨h′(t_{1}), J(t_{2})⟩dt_{1}dt_{2}

−\frac{1}{4} ∫_{0}^{1} ∫_{0}^{1} G_{P}^{2}(t_{1}, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}

−\frac{1}{2} ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} sinh(−(t_{1} + t_{2}) + s_{i+1} + s_{i})G_{P}(t_{1}, t_{2})K⟨h′(t_{1}), J(t_{2})⟩dt_{1}dt_{2}

−\frac{1}{4} ∫_{0}^{1} ∫_{0}^{1} G_{P}(t_{1}, t_{2})^{2}K⟨h(t_{1}), J(t_{2})⟩dt_{1}dt_{2},

which gives (3.5).□ Lemma 3.1.4 G_{P} converges to G uniformly on [0, 1]^{2} as |P| → 0.

Proof: Recall that G(s, t) = G_{P}(s, t) + ∑_{i=0}^{n−1} G_{i}(s, t) for (s, t) ∈ [0, 1]^{2}, where

G_{i}(s, t) = \{ \begin{matrix}0 & if (s, t) ∉ [s_{i}, s_{i+1}]^{2} \\ \frac{sinh(t−s_{i}) sinh(s_{i+1}−s)}{sinh(s_{i+1}−s_{i})} & if s_{i} ≤ t < s ≤ s_{i+1} \\ \frac{sinh(s−s_{i}) sinh(s_{i+1}−t)}{sinh(s_{i+1}−s_{i})} & if s_{i} ≤ s ≤ t ≤ s_{i+1} \end{matrix} .

If (s, t) ∉ [s_{i}, s_{i+1}]^{2} ∀i, then clearly G(s, t) = G_{P}(s, t). If (s, t) ∈ [s_{i}, s_{i+1}]^{2} for some i, then for |P| sufficiently small,

|G(s, t) − G_{P}(s, t)| = |G_{i}(s, t)|
≤ sinh(s_{i+1} − s_{i}) ≤ sinh(|P|),

since sinh is an increasing function. In any case, |G(s, t) − G_{P}(s, t)| ≤ sinh(|P|) → 0 as |P| → 0. □

The next three lemmas demonstrate the appropriate approximation of the terms

in (Δ_{P}h, J) to those in (Δh, J).

Lemma 3.1.5 For a given partition P and h, J ∈ H_{P}(𝔤), let

E(h, J, P) ≡ |∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} sinh(−(t_{1} + t_{2}) + s_{i+1} + s_{i})

·G_{P}(t_{1}, t_{2})(K⟨h′(t_{1}), J(t_{2})⟩ + K⟨h(t_{1}), J′(t_{2})⟩)dt_{1}dt_{2}| .

Then

lim_{|P|→0} sup_{∥h∥,∥J∥≤1} E(h, J, P) = 0.

Proof: First note that

E(h, J, P)

≤ ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} |\frac{sinh(s_{i+1} − t_{2})}{sinh(s_{i+1} − s_{i})} cosh(t_{1} − s_{i}) − \frac{sinh(t_{1} − s_{i})}{sinh(s_{i+1} − s_{i})} cosh(s_{I+1} − t_{2})|

·M · |K⟨h′(t_{1}), J(t_{2})⟩ + K⟨h(t_{1}), J′(t_{2})⟩| dt_{1}dt_{2},

where M is a bound for G_{P}(t_{1}, t_{2}) on [0, 1]^{2}. For small enough |P|, we have |\frac{sinh(s_{i+1}−t_{2})}{sinh(s_{i+1}−s_{i})} | and |\frac{sinh(t_{1}−s_{i})}{sinh(s_{i+1}−s_{i})} | less than 1 , and |cosh(t_{1} − s_{i})| and |cosh(s_{I+1} − t_{2})| less than 2. Thus, by use of the Cauchy-Schwartz inequality,

E(h, J, P)
≤ 4M ∫_{0}^{1} ∫_{0}^{1}(∑_{i=0}^{n−1} 1_{[s_{i},s_{i+1}]^{2}}) |K⟨h′(t_{1}), J(t_{2})⟩ + K⟨h(t_{1}), J′(t_{2})⟩| dt_{1}dt_{2}
≤ 4M(∫_{0}^{1} ∫_{0}^{1}(∑_{i=0}^{n−1} 1_{[s_{i},s_{i+1}]^{2}})^{2}dt_{1}dt_{2})^{\frac{1}{2}}
·(∫_{0}^{1} ∫_{0}^{1} |K⟨h′(t_{1}), J(t_{2})⟩ + K⟨h(t_{1}), J′(t_{2})⟩|^{2} dt_{1}dt_{2})^{\frac{1}{2}}
≤ 4M(∑_{i=0}^{n−1}(s_{i+1} − s_{i})^{2})^{\frac{1}{2}} (∫_{0}^{1} ∫_{0}^{1} 2(|K⟨h′(t_{1}), J(t_{2})⟩|^{2} + |K⟨h(t_{1}), J′(t_{2})⟩|^{2})dt_{1}dt_{2})^{\frac{1}{2}}
≤ 4\sqrt{2}M(∑_{i=0}^{n−1}(s_{i+1} − s_{i})^{2})^{\frac{1}{2}}
·(∫_{0}^{1} ∫_{0}^{1} c^{2} |h′(t_{1})|^{2} |J(t_{2})|^{2} dt_{1}dt_{2} + ∫_{0}^{1} ∫_{0}^{1} c^{2} |h(t_{1})|^{2} |J′(t_{2})|^{2} dt_{1}dt_{2})^{\frac{1}{2}}
≤ 4\sqrt{2}Mc(∑_{i=0}^{n−1}(s_{i+1} − s_{i})^{2})^{\frac{1}{2}} (2 ∥h∥^{2} ∥J∥^{2})^{\frac{1}{2}} = 8Mc(∑_{i=0}^{n−1}(s_{i+1} − s_{i})^{2})^{\frac{1}{2}} ∥h∥ ∥J∥ ,

where c is a constant depending on K⟨·, ·⟩. Therefore,

lim_{|P|→0} sup_{∥h∥,∥J∥≤1} E(h, J, P) ≤ lim_{|P|→0} 8Mc(∑_{i=0}^{n−1}(s_{i+1} − s_{i})^{2})^{\frac{1}{2}} = 0,

which proves the lemma.□

Lemma 3.1.6 For each partition P of [0, 1] there exists an ϵ(P) ≥ 0 such that ϵ(P) → 0 as |P| → 0 and for h, J ∈ H_{P}(𝔤),

|||||||

∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i}[cosh(s_{I} − t_{1} + s_{i+1} − t_{2})G_{P}(t_{1}, t_{2}) − a cosh(\frac{1}{2})]

·K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}

|||||||

≤ ϵ(P) ∥h∥ ∥J∥

Proof: Note first that cosh is uniformly continuous on [0, 1], G_{P} converges

uniformly to G on [0, 1] as |P| → 0 (by Lemma 3.1.4), and G is uniformly continuous on [0, 1]^{2}. For a given P, let

ϵ(P) ≡ max_{i∈\{1,... ,n−1\}} \{ sup_{(t_{1},t_{2})∈[s_{i},s_{i+1}]^{2}} \{|cosh(s_{I} − t_{1} + s_{i+1} − t_{2})G_{P}(t_{1}, t_{2}) − a cosh(\frac{1}{2})|\}\} .

Letting A(t_{1}, t_{2}, s_{i}, s_{i+1}) ≡ |cosh(s_{I} − t_{1} + s_{i+1} − t_{2})G_{P}(t_{1}, t_{2}) − a cosh(\frac{1}{2})|, then we see that

A(t_{1}, t_{2}, s_{i}, s_{i+1})
= |cosh(s_{I} − t_{1} + s_{i+1} − t_{2})G_{P}(t_{1}, t_{2}) − cosh(s_{I} − t_{1} + s_{i+1} − t_{2})G(t_{1}, t_{2})
+ cosh(s_{I} − t_{1} + s_{i+1} − t_{2})G(t_{1}, t_{2}) − G(t_{1}, t_{2})
+G(t_{1}, t_{2}) − G(t_{2}, t_{2})|
≤ |cosh(s_{I} − t_{1} + s_{i+1} − t_{2})| |G_{P}(t_{1}, t_{2}) − G(t_{1}, t_{2})|
+ |cosh(s_{I} − t_{1} + s_{i+1} − t_{2}) − 1| |G(t_{1}, t_{2})| + |G(t_{1}, t_{2}) − G(t_{1}, t_{2})|

where we have used the fact that a cosh(\frac{1}{2}) = aG(t_{2}, t_{2}). It is then clear that ϵ(P) → 0 as |P| → 0 by our remarks at the beginning of the proof.

We now have that

|∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i}A(t_{1}, t_{2}, s_{i}, s_{i+1})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dtw|

≤ ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} |A(t_{1}, t_{2}, s_{i}, s_{i+1})| |K⟨h′(t_{1}), J′(t_{2})⟩| dt_{1}dt_{2}
≤ ϵ(P) ∫_{[0,1]^{2}}(∑_{i=0}^{n−1} \frac{1}{sinh(s_{i+1} − s_{i})}1_{[s_{i},s_{i+1}]^{2}}(t_{1}, t_{2})) |K⟨h′(t_{1}), J′(t_{2})⟩| dt_{1}dt_{2}
≤ ϵ(P) ∫_{[0,1]^{2}}(∑_{i=0}^{n−1} \frac{1}{Δ_{i}} 1_{[s_{i},s_{i+1}]^{2}}(t_{1}, t_{2})) |K⟨h′(t_{1}), J′(t_{2})⟩| dt_{1}dt_{2},

where we have used the fact that Δ_{i} ≡ s_{i+1} − s_{i} < sinh(s_{i+1} − s_{i}) for all i (note that sinh′(x) = cosh(x) > 1 ⇒ sinh(x) > x for x > 0). Letting

δ_{P}(t_{1}, t_{2}) ≡ ∑_{i=0}^{n−1} \frac{1}{Δ_{i}} 1_{[s_{i},s_{i+1}]^{2}}(t_{1}, t_{2})

for (t_{1}, t_{2}) ∈ [0, 1], the Cauchy-Schwartz inequality gives

|∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i}A(t_{1}, t_{2}, s_{i}, s_{i+1})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}|

≤ ϵ(P) ∫_{[0,1]^{2}} δ_{P}(t_{1}, t_{2}) |h′(t_{1})| |J′(t_{2})| dt_{1}dt_{2}
= ϵ(P) ∫_{0}^{1} |h′(t_{1})| (T_{P} |J′(·)|)(t_{1})dt_{1}
≤ ϵ(P) ∥h′∥_{L^{2}(H_{P}(𝔤))} ∥T_{P} |J′(·)|∥_{L^{2}([0,1]).}

where (T_{P}f)(s) ≡ ∫_{0}^{1} δ_{P}(t, s)f(s)ds for all f ∈.L^{2}([0, 1]).

We claim that T_{P} is a bounded operator on L^{2}([0, 1]). Indeed, for f ∈.L^{2}([0, 1]),

∫_{0}^{1} |(T_{P}f)(t)|^{2} dt = ∫_{0}^{1} |∫_{0}^{1} δ_{P}(t, s)f(s)ds|^{2} dt
= ∫_{0}^{1} |∫_{0}^{1}(∑_{i=0}^{n−1} \frac{1}{Δ_{i}} 1_{[s_{i},s_{i+1}]^{2}}(t, s))f(s)ds|^{2} dt
= ∫_{0}^{1} |∑_{i=0}^{n−1} 1_{[s_{i},s_{i+1}]}(t) ∫_{s_{i}}^{s_{i+1}} f(s)\frac{1}{Δ_{i}} ds|^{2} dt
= ∫_{0}^{1} ∑_{i=0}^{n−1} 1_{[s_{i},s_{i+1}]}(t)(∫_{s_{i}}^{s_{i+1}} f(s)\frac{1}{Δ_{i}} ds)^{2}dt
= ∑_{i=0}^{n−1} Δ_{i}(∫_{s_{i}}^{s_{i+1}} f(s)\frac{1}{Δ_{i}} ds)^{2}
≤ ∑_{i=0}^{n−1} Δ_{i} ∫_{s_{i}}^{s_{i+1}} |f(s)|^{2} \frac{1}{Δ_{i}} ds
= ∫_{0}^{1} |f(s)|^{2} ds = ∥f∥_{L^{2}([0,1])} ,

where we have used Jensen’s inequality in the last inequality.

Hence, ∥T_{P}∥_{L^{2}→L^{2}} ≤ 1, and we then have

|∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i}A(t_{1}, t_{2}, s_{i}, s_{i+1})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}|

≤ ϵ(P) ∥h′∥_{L^{2}(H_{P}(𝔤))} ∥T_{P}∥_{L^{2}→L^{2}} ∥J′∥_{L^{2}(H_{P}(𝔤))} ≤ ϵ(P) ∥h∥_{H(𝔤)} ∥J∥_{H(𝔤)}

which proves the lemma.□

Lemma 3.1.7 For each partition P of [0, 1] there exists an ϵ(P) ≥ 0 such that ϵ(P) → 0 as |P| → 0 and for h, J ∈ H_{P}(𝔤),

|∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i}K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2} − ∫_{0}^{1} K⟨h′(t), J′(t)⟩dt| ≤ ϵ(P) ∥h∥ ∥J∥ . (3.11)

Proof: There are certain simplifications we can make in (3.11 ) to make calcu-

lations easier. First, it is sufficient to prove the lemma with a_{i} = \frac{1}{sinh(s_{i+1}−s_{i})} replaced by \frac{1}{Δ_{i}} = \frac{1}{s_{i+1}−s_{i}}. Indeed, by Taylor’s theorem, for |P| sufficiently small,

0 ≤ sinh(s_{i+1} − s_{i}) = s_{i+1} − s_{i} + \frac{cosh(ξ_{i})(s_{i+1} − s_{i})^{3}}{3!} ∀i,

where 0 < ξ_{i} < s_{i+1} − s_{i} and 1 < cosh(ξ_{i}) < 2. Thus, for all i,

|\frac{1}{sinh(s_{i+1} − s_{i})} − \frac{1}{s_{i+1} − s_{i}} | = \frac{\frac{cosh(ξ_{i})}{3!}(s_{i+1} − s_{i})^{3}}{(s_{i+1} − s_{i} + \frac{cosh(ξ_{i})}{3!}(s_{i+1} − s_{i})^{3})(s_{i+1} − s_{i})}
≤ s_{i+1} − s_{i} ≤ |P| .

Second, it is sufficient to replace J by h. If we define the bilinear form B by

E⟨h, J⟩ ≡ ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i}K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2} − ∫_{0}^{1} K⟨h′(t), J′(t)⟩dt,

then by the symmetry of K, E is symmetric. Since H_{P}(𝔤) is finite dimensional, then for all h ∈ H_{P}(𝔤) the linear functional E⟨h, ·⟩ : H_{P}(𝔤) → R is bounded. Thus, by the Riesz representation theorem, there exists a linear operator A : H_{P}(𝔤) → H_{P}(𝔤) such that E⟨h, J⟩ = ⟨Ah, J⟩ for all h, J ∈ H_{P}(𝔤). Since E is symmetric, then so is A. If we assume that |E⟨h, h⟩| ≤ ϵ(P) ∥h∥^{2} for all h ∈ H_{P}(𝔤), i.e., |⟨Ah, h⟩| ≤ ϵ(P) ∥h∥^{2} for all h ∈ H_{P}(𝔤), then by Lemma 2.1 in Lang [15] and by the symmetry of A,

|E⟨h, J⟩| = |⟨Ah, J⟩|
≤ ϵ(P) ∥h∥^{2} ∥J∥^{2}

for all h, J ∈ H_{P}(𝔤).

Third, it is sufficient to prove (3.11) assuming h is real and with K⟨·, ·⟩ replaced

by scalar multiplication. To see why, recall that H_{P}(𝔤) = {G(·, s_{i})A : A ∈ 𝔤, s_{i} ∈ P} = H_{P}(R)𝔤. We can choose a basis, {e_{1}, …, e_{d}} (where d = dim(𝔤)), which is orthonormal with respect to K⟨·, ·⟩, i. e., K⟨e_{i}, e_{j}⟩ = δ_{ij} for i, j ∈ {1, …, d}. We may then write h = ∑_{j=1}^{d} h_{j}e_{j}, where h_{j} ∈ H_{P}(R) for each j ∈ {1, …, d}. Assuming (3.11) is true for the scalar case, then

|∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i}K⟨h′(t_{1}), h′(t_{2})⟩dt_{1}dt_{2} − ∫_{0}^{1} K⟨h′(t), h′(t)⟩dt|

= |∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i}{∑_{j=1}^{d} h′_{j}(t_{1})h′_{j}(t_{2})}dt_{1}dt_{2} − ∫_{0}^{1}{∑_{j=1}^{d} h_{j}(t)h_{j}(t)}dt|
≤ ∑_{j=1}^{d} ϵ(P) ∥h_{j}∥_{H_{P}(R)}^{2}

= ϵ(P) ∫_{0}^{1} ∑_{j=1}^{d}{|h′_{j}(t)|^{2} + |h_{j}(t)|^{2}}dt
= ϵ(P) ∫_{0}^{1}{K⟨h′(t), h′(t)⟩ + K⟨h(t), h(t)⟩}dt
≤ ϵ(P) ∫_{0}^{1}{c |h′(t)|^{2} + c |h(t)|^{2}}dt
= cϵ(P) ∥h∥_{H_{P}(𝔤)}^{2}

where c is a constant depending on K.

By the fundamental theorem of calculus,

∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} \frac{1}{Δ_{i}} h′(t_{1}) · h′(t_{2})dt_{1}dt_{2} = ∑_{i=0}^{n−1} \frac{Δ_{i}h}{Δ_{i}} · \frac{Δ_{i}h}{Δ_{i}}Δ_{i},

where Δ_{i}h ≡ h(s_{i+1}) − h(s_{i}). We thus need to prove that for each partition P, there exists an ϵ(P) such that

|∑_{i=0}^{n−1} \frac{Δ_{i}h}{Δ_{i}} · \frac{Δ_{i}h}{Δ_{i}}Δ_{i} − ∫_{0}^{1} h′(t) · h′(t)⟩dt| ≤ ϵ(P) ∥h∥^{2} ,

where ϵ(P) → 0 as |P| → 0. But note that

∑_{i=0}^{n−1} \frac{Δ_{i}h}{Δ_{i}} · \frac{Δ_{i}h}{Δ_{i}}Δ_{i} = ∑_{i=0}^{n−1} h̃′_{i}(t) · h̃′_{i}(t)Δ_{i}
= ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} h̃′_{i}(t) · h̃′_{i}(t)dt,

where for each i, h̃_{i} is the line segment with h̃_{i}(s_{i}) = h(s_{i}) and h̃_{i}(s_{i+1}) = h(s_{i+1}).

For each i, consider the integral ∫_{s_{i}}^{s_{i+1}}{|h̃′_{i}(t)|^{2} − |h′(t)|^{2}}dt. Recall that h ∈

H_{P}(R) is of the form h(t) = ∑_{j=0}^{n−1} c_{j}G(t, s_{j}) and can thus be expressed as h(t) = ae^{t} +be^{−t} when restricted to [s_{i}, s_{i+1}]. In light of this, h is then uniquely determined by its values at the endpoints: α ≡ h(s_{i}) and β ≡ h(s_{i+1}). Since integration is translation invariant, then, for convenience in calculations, we may write

∫_{s_{i}}^{s_{i+1}}{|h̃′_{i}(t)|^{2} − |h′(t)|^{2}}dt = ∫_{−δ}^{δ}{|k̃′(t)|^{2} − |k′(t)|^{2}}dt

where k is of the form k = ce^{t} + de^{−t} with k(−δ) = α = h(s_{i}) and k(δ) = β = h(s_{i+1}), and k̃ is a line with k̃(−δ) = α and k̃(δ) = β. One can easily verify that

k(t) = \frac{1}{sinh(2δ)}(β sinh(δ + t) + α sinh(δ − t)),
k̃(t) = α + \frac{β − α}{2δ}(t + δ) = \frac{1}{2δ} (β(δ + t) + α(δ − t)),
k′(t) = \frac{1}{sinh(2δ)}(β cosh(δ + t) − α cosh(δ − t)),
k̃′(t) = \frac{β − α}{2δ}.

Letting b ≡ sinh(2δ), we then have the following:

∫_{−δ}^{δ} |k′(t)|^{2} dt

= \frac{1}{b^{2}} ∫_{−δ}^{δ}(β cosh(δ + t) − α cosh(δ − t))^{2}dt
= \frac{1}{b^{2}} ∫_{−δ}^{δ}(β(cosh(δ) cosh(t) + sinh(δ) sinh(t))

−α(cosh(δ) cosh(t) − sinh(δ) sinh(t)))^{2}dt

= \frac{1}{b^{2}} ∫_{−δ}^{δ}((β − α) cosh(δ) cosh(t) + (β + α) sinh(δ) sinh(t))^{2}dt
= \frac{1}{b^{2}} ∫_{−δ}^{δ}\{(β − α)^{2} cosh^{2}(δ) cosh^{2}(t) + (β + α)^{2} sinh^{2}(δ) sinh^{2}(t)

+2(β − α)(β + α) cosh(δ) cosh(t) sinh(δ) sinh(t)}dt

= \frac{1}{b^{2}} ∫_{−δ}^{δ}\{(β − α)^{2} cosh^{2}(δ) cosh^{2}(t) + (β + α)^{2} sinh^{2}(δ) sinh^{2}(t)\}dt
= \frac{1}{b^{2}} ∫_{−δ}^{δ}\{(β − α)^{2} cosh^{2}(δ)\frac{1}{2}(cosh(2t) + 1) + (β + α)^{2} sinh^{2}(δ)\frac{1}{2}(cosh(2t) − 1)\}dt
= \frac{1}{b^{2}} \{(β − α)^{2} cosh^{2}(δ)\frac{1}{2}(sinh(2t) + 2δ) + (β + α)^{2} sinh^{2}(δ)\frac{1}{2}(sinh(2t) − 2δ)\}.

Noting that ∫_{−δ}^{δ} |k̃′(t)|^{2} dt = ∫_{−δ}^{δ} \frac{(β−α)^{2}}{4δ^{2}} dt = \frac{(β−α)^{2}}{2δ}, we then have

|∫_{−δ}^{δ}{|k̃′(t)|^{2} − |k′(t)|^{2}}dt|
= |(β − α)^{2}\{\frac{1}{2δ} − \frac{1}{sinh^{2}(2δ)} cosh^{2}(δ)\frac{1}{2}(sinh(2δ) + 2δ)\}

−(β + α)^{2} \frac{1}{sinh^{2}(2δ)} sinh^{2}(δ)\frac{1}{2}(sinh(2δ) − 2δ)|

≤ (β − α)^{2} \frac{1}{sinh(2δ)} |\frac{sinh(2δ)}{2δ} − cosh^{2}(δ)\frac{1}{2}(1 + \frac{2δ}{sinh(2δ)})|

+(β + α)^{2} \frac{1}{sinh(2δ)} sinh^{2}(δ)\frac{1}{2} |1 − \frac{2δ}{sinh(2δ)}|

≤ \frac{1}{b}\{(β − α)^{2} + sinh^{2}(δ)(β + α)^{2} \frac{1}{2}\}ϵ(δ)
≤ \frac{1}{b}\{(β − α)^{2} + (cosh^{2}(δ) − 1)(β^{2} + α^{2})\}ϵ(δ)
≤ \frac{1}{b}\{(β − α)^{2} + (cosh(2δ) − 1)(β^{2} + α^{2})\}ϵ(δ),  (3.12)

where ϵ(δ) ≡ max\{|\frac{sinh(2δ)}{2δ} − cosh^{2}(δ)\frac{1}{2}(1 + \frac{2δ}{sinh(2δ)})| , |1 − \frac{2δ}{sinh(2δ)} |\}, and, for the last inequality, we used the fact that cosh^{2}(δ) ≤ cosh(2δ) since cosh(2δ) = cosh^{2}(δ)+sinh^{2}(δ) . It is easy to check that ϵ(δ) → 0 as δ → 0.

We now compare the expression in (3.12) to (k, k)_{δ} ≡ ∫_{−δ}^{δ}{|k′(t)|^{2} + |k(t)|^{2}}dt.

Since, in the interval (−δ, δ), k has the property that k′′ = k, we can use integration by parts to get

(k, k)_{δ} = k′k|_{−δ}^{δ} − ∫_{−δ}^{δ} |k(t)|^{2} dt + ∫_{−δ}^{δ} |k(t)|^{2} dt
= \frac{1}{b^{2}} (β cosh(δ + t) − α cosh(δ − t))(β sinh(δ + t) + α sinh(δ − t))|_{−δ}^{δ}
= \frac{1}{b^{2}} \{(β cosh(2δ) − α)β sinh(2δ) − (β − α cosh(2δ))α sinh(2δ)\}
= \frac{1}{b}\{cosh(2δ)(α^{2} + β^{2}) − 2αβ\}
= \frac{1}{b}\{(β − α)^{2} + (cosh(2δ) − 1)(α^{2} + β^{2})\}.

Therefore, |∫_{−δ}^{δ}{|k̃′(t)|^{2} − |k′(t)|^{2}}dt| ≤ ϵ(δ)(k, k)_{δ}.

Returning to our original notation and using translation invariance of integra-

tion, we have

|∫_{s_{i}}^{s_{i+1}}{|h̃′(t)|^{2} − |h(t)|^{2}}dt| ≤ ϵ(s_{i+1} − s_{i})(h, h)_{i},

where (h, h)_{i} = ∫_{s_{i}}^{s_{i+1}}{|h′(t)|^{2} + |h(t)|^{2}}dt. Thus,

|∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}}{|h̃′(t)|^{2} − |h(t)|^{2}}dt| ≤ ∑_{i=0}^{n−1} ϵ(s_{i+1} − s_{i})(h, h)_{i}
≤ ϵ(|P|)(h, h) = ϵ(|P|) ∥h∥^{2} .

This completes the proof of the lemma. □

We are now ready to prove (3.3). By Lemma 3.4 and Lemma 3.5 we get

|(Δh, J) − (Δ_{P}h, J)|

≤ |∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} cosh(s_{i} − t_{1} + s_{i+1} − t_{2})G_{P}(t_{1}, t_{2})K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}

− ∫_{0}^{1} a cosh(\frac{1}{2})K⟨h′(t), J′(t)⟩dt|

+ |\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1}(G_{P}^{2}(t_{1}, t_{2}) − G^{2}(t_{1}, t_{2}))K ⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}|

+ |\frac{1}{4} ∫_{0}^{1} ∫_{0}^{1}(G_{P}^{2}(t_{1}, t_{2}) − G^{2}(t_{1}, t_{2}))K⟨h(t_{1}), J(t_{2})⟩dt_{1}dt_{2}|

+ |∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i} sinh(−(t_{2} + t_{1}) + s_{i+1} + s_{i})

.G_{P}(t_{1}, t_{2})(K⟨h′(t_{1}), J(t_{2})⟩ + K⟨h(t_{1}), J′(t_{2})⟩)dt_{1}dt_{2}|
≤ |∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i}[cosh(s_{I} − t_{1} + s_{i+1} − t_{2})G_{P}(t_{1}, t_{2}) − a cosh(\frac{1}{2})]

·K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2}|

+a cosh(\frac{1}{2}) |∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} ∫_{s_{i}}^{s_{i+1}} a_{i}K⟨h′(t_{1}), J′(t_{2})⟩dt_{1}dt_{2} − ∫_{0}^{1} K⟨h′(t), J′(t)⟩dt|

+\frac{1}{2} ∫_{0}^{1} ∫_{0}^{1} |G_{P}^{2}(t_{1}, t_{2}) − G^{2}(t_{1}, t_{2})| c |h′(t_{1})| |J′(t_{2})| dt_{1}dt_{2}

+\frac{1}{4} ∫_{0}^{1} ∫_{0}^{1} |G_{P}^{2}(t_{1}, t_{2}) − G^{2}(t_{1}, t_{2})| c |h(t_{1})| |J(t_{2})| dt_{1}dt_{2}

+E(h, J, P)

≤ {ϵ_{1}(P) + ϵ_{2}(P) + ϵ_{3}(P) + ϵ_{4}(P)} ∥h∥ ∥J∥

where ϵ_{1}(P) is from Lemma 3.1.6, ϵ_{2}(P) is from Lemma 3.1.7,

ϵ_{3}(P) ≡ \frac{3}{4}c sup_{(t_{1},t_{2})∈[0,1]^{2}}\{G_{P}(t_{1}, t_{2}) − G(t_{1}, t_{2})\}, E(h, J, P)

(c depends on K), E(h, J, P) is from Lemma 3.1.5, and ϵ_{4}(P) ≡ sup_{∥k∥,∥l∥≤1}{E(k, l, P)}. Noting that ϵ_{3}(P) → 0 as |P| → 0 (by Lemma 3.1.4, then from the above lemmas it is clear that ϵ(P) ≡ {ϵ_{1}(P) + ϵ_{2}(P) + ϵ_{3}(P) + ϵ_{4}(P)} → 0 as |P| → 0. Using this definition of ϵ(P) in (3.3), we have now completed the proof for Theorem 3.1.1. □

3.2 Convergence of Finite Dimensional Ricci Tensor

Theorem 3.2.1 ∥P_{P}(Ric − Ric_{P})P_{P}∥_{op} → 0 as |P| → 0.

Proof: Our goal is to find, for each partition P, an ϵ(P) such that ϵ(P) → 0 as

|P| → 0 and, for each h ∈ H_{P}(𝔤),

|((Ric − Ric_{P})h, h)| ≡ |Ric⟨h, h⟩ − Ric_{P}⟨h, h⟩|
≤ ϵ(P) ∥h∥^{2} .

We first note that for a good orthonormal basis, S, of H(𝔤), we have

Ric ⟨h, h⟩ = ∑_{k∈S}(R ⟨h, k⟩ k, h)
= ∑_{k∈S}(∇_{h}∇_{k}k − ∇_{k}∇_{h}k − ∇_{[h,k]}k, h)
= ∑_{k∈S} −(∇_{k}∇_{h}k, h) − (∇_{[h,k]}k, h)
= ∑_{k∈S}(∇_{h}k, ∇_{k}h) − (∇_{[h,k]}k, h)
= ∑_{k∈S}(∇_{k}h + [h, k], ∇_{k}h) − (∇_{[h,k]}k, h)
= ∑_{k∈S} −(∇_{k}∇_{k}h, h) − (∇_{k}[h, k] + ∇_{[h,k]}k, h)
= −(Δh, h) − ∑_{k∈S}(∇_{k}[h, k] + ∇_{[h,k]}k, h).

Similarly,

Ric_{P} ⟨h, h⟩

= −(Δ_{P}h, h) − ∑_{k∈S_{P}}(∇_{k}[h, k]_{P} + ∇_{[h,k]_{P}} k, h)
= −(Δ_{P}h, h) − ∑_{k∈S_{P}}(∇_{k}[h, k] + ∇_{[h,k]}k, h) + ∑_{k∈S_{P}}(∇_{k}(Q_{P}[h, k]) + ∇_{Q_{P}[h,k]}k, h)
= −(Δ_{P}h, h) − ∑_{k∈S_{P}}(∇_{k}[h, k] + ∇_{[h,k]}k, h) + ∑_{k∈S_{P}}(2∇_{k}(Q_{P}[h, k]) + [Q_{P}[h, k], k], h)

where Q_{P} = I − P_{P} and I = identity on H(𝔤).

Let us compare Ric ⟨h, h⟩ and Ric_{P} ⟨h, h⟩ . By Theorem 3.1.1, we know there

exists an ϵ_{1}(P) such that |(Δh, h) − (Δ_{P}h, h)| ≤ ϵ_{1}(P) ∥h∥^{2} with ϵ_{1}(P) → 0 as |P| → 0 . Next, note that for k ∈ S

(∇_{k}[h, k] + ∇_{[h,k]}k, h)

= (− ∫_{0}^{1} F (·, t)[k, [h, k]′](t)dt + \frac{1}{2} ∫_{0}^{1} G(·, t)[k, [h, k]](t)dt

− ∫_{0}^{1} F (·, t)[[h, k], k′](t)dt + \frac{1}{2} ∫_{0}^{1} G(·, t)[[h, k], k](t)dt, h)

= (− ∫_{0}^{1} F (·, t){[k, [h, k]′](t) + [[h, k], k′](t)}dt, h)
= (− ∫_{0}^{1} F (·, t){[k, [h′, k]](t) + [k, [h, k′]](t) + [[h, k], k′](t)}dt, h)
= (− ∫_{0}^{1} F (·, t){[k, [h′, k]](t) − [k′, [k, h]](t) − [h, [k′, k]](t) + [[h, k], k′](t)}dt, h)
= (− ∫_{0}^{1} F (·, t)[k, [h′, k]](t)dt, h)
= ∫_{0}^{1}{⟨− ∫_{0}^{1} G_{ss}(s, t)[k, [h′, k]](t)dt, h′(s)⟩

+⟨− ∫_{0}^{1} F (s, t)[k, [h′, k]](t)dt, h(s)⟩}ds

= ∫_{0}^{1}{⟨− ∫_{0}^{1}(δ(t − s) − G(s, t))[k, [h′, k]](t)dt, h′(s)⟩

+⟨− ∫_{0}^{1} G_{s}(s, t)[k, [h′, k]](t)dt, h(s)⟩}ds

= ∫_{0}^{1}⟨[k, [h′, k]](s), h′(s)⟩ds

where we have used integration by parts for the last equation and the fact that S is a “good” basis (i.e., k ∈ S ⟹ [k′, k] = 0 ) for the fifth equation.

Thus,

∑_{k∈S}(∇_{k}[h, k] + ∇_{[h,k]}k, h) = ∑_{k∈S} ∫_{0}^{1}⟨[k, h′](s), [k, h′](s)⟩ds
= ∫_{0}^{1} G(s, s)K⟨h′(s), h′(s)⟩ds.

Similarly,

∑_{k∈S_{P}}(∇_{k}[h, k] + ∇_{[h,k]}k, h) = ∫_{0}^{1} G_{P}(s, s)K⟨h′(s), h′(s)⟩ds.

It follows then that

|∑_{k∈S}(∇_{k}[h, k] + ∇_{[h,k]}k, h) − ∑_{k∈S_{P}}(∇_{k}[h, k] + ∇_{[h,k]}k, h)|

≤ ∫_{0}^{1} |G(s, s) − G_{P}(s, s)| K⟨h′(s), h′(s)⟩ds
≤ ϵ_{2}(P) ∫_{0}^{1} |⟨h′(s), h′(s)⟩| ds
≤ ϵ_{2}(P) ∥h∥^{2}

where ϵ_{2}(P) = (sup_{s∈[0,1]} |G(s, s) − G_{P}(s, s)|) · C and C depends on K⟨·, ·⟩ . Since we know by Lemma 3.1.4 that G_{P} → 0 uniformly as |P| → 0 , then ϵ_{2}(P) → 0 as |P| → 0 .

We next claim that for each partition, P , there exists an ϵ_{3}(P) such that

ϵ_{3}(P) → 0 as |P| → 0 and for h ∈ H_{P}(𝔤),

| ∑_{k∈S_{P}}([Q_{P}[h, k], k], h)| ≤ ϵ_{3}(P) ∥h∥^{2} .

Since h ∈ H_{P}(𝔤), for an interval, [s_{i}, s_{i+1}], of the partition P, h must be of the form h(t) = ce^{t} + de^{−t} for all t ∈ [s_{i}, s_{i+1}] . It is easy to check that in fact

h(t) = h(s_{i}) cosh(t− s_{i}) + \frac{Δ_{i}h}{η_{i}} sinh(t− s_{i}) ∀t ∈ [s_{i}, s_{i+1}],

where s_{i}≡ \frac{s_{i}+s_{i+1}}{2} , Δ_{i}h ≡ h(s_{i+1}) − h(s_{i}) , and η_{i} ≡ 2 sinh(\frac{s_{i+1}−s_{i}}{2} ) .

Letting Δ_{i} ≡ s_{i+1} − s_{i} , we may then write, for t ∈ [s_{i}, s_{i+1}] , [h, k](t)

= [h(s_{i}) cosh(t− s_{i}) + \frac{Δ_{i}h}{η_{i}} sinh(t− s_{i}), k(s_{i}) cosh(t− s_{i}) + \frac{Δ_{i}k}{η_{i}} sinh(t− s_{i})
= [h(s_{i}), k(s_{i})] cosh^{2}(t− s_{i}) + \frac{1}{η_{i}} ([Δ_{i}h, k(s_{i})] + [h(s_{i}), Δ_{i}k]) cosh(t− s_{i}) sinh(t− s_{i})

+\frac{1}{η_{i}^{2}} [Δ_{i}h, Δ_{i}k] sinh^{2}(t− s_{i}),

and

Δ_{i}[h, k] = ([Δ_{i}h, k(s_{i})] + [h(s_{i}), Δ_{i}k]) cosh(\frac{Δ_{i}}{2} ).

Using Proposition 3.0.3 we then have

P_{P}[h, k](t) = \frac{[h, k](s_{i}) + [h, k](s_{i+1})}{ς_{i}} cosh(t− s_{i}) + \frac{Δ_{i}[h, k]}{η_{i}} sinh(t− s_{i})
= \frac{\{2[h(s_{i}), k(s_{i})] cosh^{2}(\frac{Δ_{i}}{2} ) + \frac{2}{η_{i}^{2}} [Δ_{i}h, Δ_{i}k] sinh^{2}(\frac{Δ_{i}}{2} )\}}{ς_{i}} cosh(t− s_{i})

+\frac{2}{η_{i}^{2}} ([Δ_{i}h, k(s_{i})] + [h(s_{i}), Δ_{i}k]) cosh(\frac{Δ_{i}}{2} ) sinh(\frac{Δ_{i}}{2} ) sinh(t− s_{i})

= \{[h(s_{i}), k(s_{i})] cosh(\frac{Δ_{i}}{2} ) + \frac{1}{2ς_{i}} [Δ_{i}h, Δ_{i}k]\} cosh(t− s_{i})

+\frac{1}{η_{i}} ([Δ_{i}h, k(s_{i})] + [h(s_{i}), Δ_{i}k]) cosh(\frac{Δ_{i}}{2} ) sinh(t− s_{i}),

where ς_{i} = 2 cosh(\frac{Δ_{i}}{2} ). Thus, for t ∈ [s_{i}, s_{i+1}] ,

Q_{P}[h, k](t)
= [h, k](t) − P_{P}[h, k](t)
= [h(s_{i}), k(s_{i})](cosh^{2}(t− s_{i}) − cosh(\frac{Δ_{i}}{2} ) cosh(t− s_{i}))

+([\frac{Δ_{i}h}{η_{i}}, k(s_{i})] + [h(s_{i}), \frac{Δ_{i}k}{η_{i}}])(cosh(t− s_{i}) − cosh(\frac{Δ_{i}}{2} )) sinh(t− s_{i})

+[\frac{Δ_{i}h}{η_{i}}, \frac{Δ_{i}k}{η_{i}}](sinh^{2}(t− s_{i}) − \frac{1}{2ς_{i}} η_{i}^{2} cosh(t− s_{i})).  (3.13)

Next note that since Q_{P}[h, k] ∈ Ker(π̃_{P}), then

[Q_{P}[h, k], k](s_{i}) = [Q_{P}[h, k](s_{i}), k(s_{i})] = 0

for all s_{i} ∈ P. Thus, [Q_{P}[h, k], k] ∈ Ker(π̃_{P}) implying that

∑_{k∈S_{P}}([Q_{P}[h, k], k], h) = 0

since h ∈ H_{P}(𝔤).

We now prove some general inequalities that will be useful for the remainder

of our proof of Theorem 3.2.1.

Lemma 3.2.2 Suppose f : [0, 1] → R is a positive function such that \frac{f(x)}{x} is increasing and f(x) = o(x), where we are using the notation o(x) to mean a function with the property that \frac{|f(x)|}{|x|} → 0 as |x| → 0. Then for each partition, P = \{0 = s_{0} < s_{1} < ⋯ < s_{n−1} < 1} of [0, 1], ∃ ϵ(P), with ϵ(P) → 0 as |P| → 0, such that for all h ∈ H_{P}(𝔤)

∑_{i=0}^{n−1} |K⟨h(s_{i}), h(s_{i})⟩f(Δ_{i})| ≤ ϵ(P) ∥h∥^{2}

and

∑_{i=0}^{n−1} |K⟨h(s_{i}), Δ_{i}h⟩f(Δ_{i})| ≤ ϵ(P) ∥h∥^{2} .

Proof:

∑_{i=0}^{n−1} |K⟨h(s_{i}), h(s_{i})⟩f(Δ_{i})| ≤ ∑_{i=0}^{n−1} Δ_{i} |K⟨h(s_{i}), h(s_{i})⟩| \frac{f(Δ_{i})}{Δ_{i}}
≤ (∑_{i=0}^{n−1} Δ_{i}) (sup_{i} |K⟨h(s_{i}), h(s_{i})⟩|) (sup_{i} \frac{f(Δ_{i})}{Δ_{i}})
≤ ( sup_{s∈[0,1]} c |h(s)|^{2}) \frac{f(|P|)}{|P|}
≤ ϵ(P) ∥h∥^{2}  (3.14)

where c depends on K and ϵ(P) ≡ 2c\frac{f(|P|)}{|P|}.

To see why the last inequality holds in (3.14), note that for s ∈ [0, 1] , h(s) =

h(t) + ∫_{t}^{s} h′(x)dx ∀t ∈ [0, 1] . Thus

h(s) = ∫_{0}^{1} h(s)dt
= ∫_{0}^{1} h(t)dt + ∫_{0}^{1} (∫_{t}^{s} h′(x)dx) dt

which implies

|h(s)|^{2} ≤ (∫_{0}^{1} |h(t)| dt + ∫_{0}^{1} (∫_{t}^{s} |h′(x)| dx) dt)^{2}
≤ 2 ((∫_{0}^{1} |h(t)| dt)^{2} + (∫_{0}^{1} |h′(x)| dx)^{2})
≤ 2 (∫_{0}^{1} |h(t)|^{2} dt + ∫_{0}^{1} |h′(x)|^{2} dx)
= 2 ∥h∥^{2}

where we have used the Cauchy Schwartz inequality in the second line and Jensen’s inequality in the third line.

The second inequality in the statement of the Lemma follows by the same

reasoning used for the first. □

Lemma 3.2.3 For h ∈ H_{P}(𝔤),

∑_{i=0}^{n−1} |K⟨Δ_{i}h, Δ_{i}h⟩| ≤ ϵ(P) ∥h∥^{2}

where ϵ(P) → 0 as |P| → 0.

Proof: Using Jensen’s inequality, we get

∑_{i=0}^{n−1} |K⟨Δ_{i}h, Δ_{i}h⟩| ≤ c ∑_{i=0}^{n−1} Δ_{i}^{2} |\frac{Δ_{i}h}{Δ_{i}} |^{2}
= c ∑_{i=0}^{n−1} Δ_{i}^{2} |∫_{s_{i}}^{s_{i+1}} h′(s)\frac{1}{Δ_{i}} ds|^{2}
≤ c ∑_{i=0}^{n−1} Δ_{i}^{2} ∫_{s_{i}}^{s_{i+1}} |h′(s)|^{2} \frac{1}{Δ_{i}} ds
≤ c |P| ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}} |h′(s)|^{2} ds
≤ c |P| ∥h∥^{2} ,

where c depends on K. The inequality follows by letting ϵ(P) ≡ c |P|. □

We now need to consider the final term in Ric_{P}⟨h, h⟩ :

∑_{k∈S_{P}}(2∇_{k}(Q_{P}[h, k]), h)

= ∑_{k∈S_{P}} 2 (− ∫_{0}^{1} F (·, t)[k, (Q_{P}[h, k])′](t)dt + \frac{1}{2} ∫_{0}^{1} G(·, t)[k, Q_{P}[h, k]](t)dt, h) (3.15)

Using our previous representation of Q_{P}[h, k] in Equation (3.13), we can split

the summand in (3.15) into two inner products as follows. The first inner product is

(− ∫_{0}^{1} F (·, t)[k, (Q_{P}[h, k])′](t)dt, h)

= ∫_{0}^{1} {⟨− ∫_{0}^{1} G_{ss}(s, t)[k, (Q_{P}[h, k])′](t)dt, h′(s)⟩

+ ⟨− ∫_{0}^{1} G_{s}(s, t)[k, (Q_{P}[h, k])′](t)dt, h(s)⟩} ds

= ∫_{0}^{1} ⟨[k(s), (Q_{P}[h, k])′(s)], h′(s)⟩ ds
= ∫_{0}^{1} ⟨(Q_{P}[h, k])′(s), [h′, k](s)⟩ ds
= ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}}⟨[h(s_{i}), k(s_{i})](2C_{i}(s)S_{i}(s) − cosh(\frac{Δ_{i}}{2} )S_{i}(s))

+\frac{1}{η_{i}} ([Δ_{i}h, k(s_{i})] + [h(s_{i}), Δ_{i}k])(S_{i}^{2}(s) + (C_{i}(s) − cosh(\frac{Δ_{i}}{2} )))C_{i}(s))

+\frac{1}{η_{i}^{2}} [Δ_{i}h, Δ_{i}k](2S_{i}(s)C_{i}(s) − \frac{1}{2ς_{i}} η_{i}^{2}S_{i}(s)),

[h(s_{i}), k(s_{i})]C_{i}(s)S_{i}(s)

+\frac{1}{η_{i}} ([Δ_{i}h, k(s_{i})]C_{i}^{2}(s) + [h(s_{i}), Δ_{i}k]S_{i}^{2}(s)) + \frac{1}{η_{i}^{2}} [Δ_{i}h, Δ_{i}k]C_{i}(s)S_{i}(s)⟩ds,(3.16)

where C_{i}(s) ≡ cosh(s− s_{i}) and S_{i}(s) ≡ sinh(s− s_{i}). For the second inner product in (3.15) we have

(∫_{0}^{1} G(·, t)[k, Q_{P}[h, k]](t)dt, h)

= ∫_{0}^{1}⟨[k, Q_{P}[h, k]](s), h(s)⟩ds
= ∫_{0}^{1}⟨Q_{P}[h, k](s), [h, k](s)⟩ds
= ∑_{i=0}^{n−1} ∫_{s_{i}}^{s_{i+1}}⟨[h(s_{i}), k(s_{i})](C_{i}^{2}(s) − cosh(\frac{Δ_{i}}{2} )C_{i}(s))

+\frac{1}{η_{i}} ([Δ_{i}h, k(s_{i})] + [h(s_{i}), Δ_{i}k])(C_{i}(s) − cosh(\frac{Δ_{i}}{2} ))S_{i}(s)

+\frac{1}{η_{i}^{2}} [Δ_{i}h, Δ_{i}k](S_{i}^{2}(s) − \frac{1}{2ς_{i}} η_{i}^{2}C_{i}(s)),

[h(s_{i}), k(s_{i})]C_{i}^{2}(s)

+\frac{1}{η_{i}} ([Δ_{i}h, k(s_{i})] + [h(s_{i}), Δ_{i}k])C_{i}(s)S_{i}(s) + \frac{1}{η_{i}^{2}} [Δ_{i}h, Δ_{i}k]S_{i}^{2}(s)⟩ds. (3.17)

Before combining the above two results to get an expression for (3.15), we make

the following definitions for brevity:

A_{1}^{i}(s) = 4C_{i}(s)S_{i}(s) − 2 cosh(\frac{Δ_{i}}{2} )S_{i}(s)
A_{2}^{i}(s) = 2S_{i}^{2}(s) + (2C_{i}(s) − cosh(\frac{Δ_{i}}{2} ))C_{i}(s)
A_{3}^{i}(s) = 4C_{i}(s)S_{i}(s) − \frac{1}{ς_{i}} η_{i}^{2}S_{i}(s)
A_{4}^{i}(s) = C_{i}(s)S_{i}(s)
A_{5}^{i}(s) = C_{i}^{2}(s)
A_{6}^{i}(s) = S_{i}^{2}(s)
A_{7}^{i}(s) = C_{i}(s)S_{i}(s)
A_{8}^{i}(s) = C_{i}^{2}(s) − cosh(\frac{Δ_{i}}{2} )C_{i}(s)
A_{9}^{i}(s) = (C_{i}(s) − cosh(\frac{Δ_{i}}{2} ))S_{i}(s)
A_{10}^{i}(s) = S_{i}^{2}(s) − \frac{1}{2ς_{i}} η_{i}^{2}C_{i}(s)
A_{11}^{i}(s) = C_{i}^{2}(s)
A_{12}^{i}(s) = C_{i}(s)S_{i}(s)
A_{13}^{i}(s) = C_{i}(s)S_{i}(s)
A_{14}^{i}(s) = S_{i}^{2}(s)

Upon expanding the two expressions in (3.16) and (3.17 ), adding them together,

and then summing over the basis S_{P}, we get the following inequality for (3.15):

| ∑_{k∈S_{P}}(2∇_{k}(Q_{P}[h, k]), h)|

≤ ∑_{i=0}^{n−1} {|K⟨h(s_{i}), h(s_{i})⟩G_{P}(s_{i}, s_{i}) ∫_{s_{i}}^{s_{i+1}}{A_{1}^{i}(s)A_{4}^{i}(s) + A_{8}^{i}(s)A_{11}^{i}(s)}ds|

+ |K⟨h(s_{i}), Δ_{i}h⟩G_{P}(s_{i}, s_{i})\frac{1}{η_{i}} ∫_{s_{i}}^{s_{i+1}} \{A_{1}^{i}(s)A_{5}^{i}(s) + A_{8}^{i}(s)A_{12}^{i}(s)\} ds|

+ |K⟨h(s_{i}), h(s_{i})⟩(G_{P}(s_{i+1}, s_{i}) − G_{P}(s_{i}, s_{i}))\frac{1}{η_{i}} ∫_{s_{i}}^{s_{i+1}} \{A_{1}^{i}(s)A_{6}^{i}(s) + A_{8}^{i}(s)A_{13}^{i}(s)\} ds|

+ |K⟨h(s_{i}), Δ_{i}h⟩(G_{P}(s_{i+1}, s_{i}) − G_{P}(s_{i}, s_{i}))\frac{1}{η_{i}} ∫_{s_{i}}^{s_{i+1}} \{A_{1}^{i}(s)A_{7}^{i}(s) + A_{8}^{i}(s)A_{14}^{i}(s)\} ds|

+ |K⟨Δ_{i}h, h(s_{i})⟩G_{P}(s_{i}, s_{i})\frac{1}{η_{i}} ∫_{s_{i}}^{s_{i+1}}\{A_{2}^{i}(s)A_{4}^{i}(s) + A_{9}^{i}(s)A_{11}^{i}(s)\}ds|

+ |K⟨Δ_{i}h, Δ_{i}h⟩G_{P}(s_{i}, s_{i})\frac{1}{η_{i}^{2}} ∫_{s_{i}}^{s_{i+1}}\{A_{2}^{i}(s)A_{5}^{i}(s) + A_{9}^{i}(s)A_{12}^{i}(s)\}ds|

+ |K⟨Δ_{i}h, h(s_{i})⟩(G_{P}(s_{i+1}, s_{i}) − G_{P}(s_{i}, s_{i}))\frac{1}{η_{i}^{2}} ∫_{s_{i}}^{s_{i+1}} \{A_{2}^{i}(s)A_{6}^{i}(s) + A_{9}^{i}(s)A_{13}^{i}(s)\} ds|

+ |K⟨Δ_{i}h, Δ_{i}h⟩(G_{P}(s_{i+1}, s_{i}) − G_{P}(s_{i}, s_{i}))\frac{1}{η_{i}^{3}} ∫_{s_{i}}^{s_{i+1}} \{A_{2}^{i}(s)A_{7}^{i}(s) + A_{9}^{i}(s)A_{14}^{i}(s)\} ds|

+ |K⟨h(s_{i}), h(s_{i})⟩(G_{P}(s_{i+1}, s_{i}) − G_{P}(s_{i}, s_{i}))\frac{1}{η_{i}} ∫_{s_{i}}^{s_{i+1}} \{A_{2}^{i}(s)A_{4}^{i}(s) + A_{9}^{i}(s)A_{11}^{i}(s)\} ds|

+ |K⟨h(s_{i}), Δ_{i}h⟩(G_{P}(s_{i+1}, s_{i}) − G_{P}(s_{i}, s_{i}))\frac{1}{η_{i}^{2}} ∫_{s_{i}}^{s_{i+1}} \{A_{2}^{i}(s)A_{5}^{i}(s) + A_{9}^{i}(s)A_{12}^{i}(s)\} ds|

+ |K⟨h(s_{i}), h(s_{i})⟩((G_{P}(s_{i+1}, s_{i+1}) − G_{P}(s_{i}, s_{i+1})) − (G_{P}(s_{i+1}, s_{i}) − G_{P}(s_{i}, s_{i}))

·\frac{1}{η_{i}^{2}} ∫_{s_{i}}^{s_{i+1}}\{A_{2}^{i}(s)A_{6}^{i}(s) + A_{9}^{i}(s)A_{13}^{i}(s)\}ds|

+ |K⟨h(s_{i}), Δ_{i}h⟩((G_{P}(s_{i+1}, s_{i+1}) − G_{P}(s_{i}, s_{i+1})) − (G_{P}(s_{i+1}, s_{I}) − G_{P}(s_{i}, s_{i}))

·\frac{1}{η_{i}^{3}} ∫_{s_{i}}^{s_{i+1}}\{A_{2}^{i}(s)A_{7}^{i}(s) + A_{9}^{i}(s)A_{14}^{i}(s)\}ds|

+ |K⟨Δ_{i}h, h(s_{i})⟩(G_{P}(s_{i+1}, s_{i}) − G_{P}(s_{i}, s_{i}))\frac{1}{η_{i}^{2}} ∫_{s_{i}}^{s_{i+1}} \{A_{3}^{i}(s)A_{4}^{i}(s) + A_{10}^{i}(s)A_{11}^{i}(s)\} ds|

+ |K⟨Δ_{i}h, Δ_{i}h⟩(G_{P}(s_{i+1}, s_{i}) − G_{P}(s_{i}, s_{i}))\frac{1}{η_{i}^{3}} ∫_{s_{i}}^{s_{i+1}} \{A_{3}^{i}(s)A_{5}^{i}(s) + A_{10}^{i}(s)A_{12}^{i}(s)\} ds|

+ |K⟨Δ_{i}h, h(s_{i})⟩((G _{P}(s_{i+1}, s_{i+1}) − G_{P}(s_{i}, s_{i+1})) − (G_{P}(s_{i+1}, s_{i}) − G_{P}(s_{i}, s_{i}))

·\frac{1}{η_{i}^{3}} ∫_{s_{i}}^{s_{i+1}}\{A_{3}^{i}(s)A_{6}^{i}(s) + A_{10}^{i}(s)A_{13}^{i}(s)\}ds|

+ |K⟨Δ_{i}h, Δ_{i}h⟩((G_{P}(s_{i+1}, s_{i+1}) − G_{P}(s_{i}, s_{i+1})) − (G_{P}(s_{i+1}, s_{i}) − G_{P}(s_{i}, s_{i}))

·\frac{1}{η_{i}^{4}} ∫_{s_{i}}^{s_{i+1}}\{A_{3}^{i}(s)A_{7}^{i}(s) + A_{10}^{i}(s)A_{14}^{i}(s)\}ds|\}

For each i ∈ {0, 1, …, n−1}, let us label the above sixteen terms by B_{1}^{i}, …, B_{16}^{i},

respectively, so that the inequality can be written more briefly as follows:

| ∑_{k∈S_{P}}(2∇_{k}(Q_{P}[h, k]), h)| ≤ ∑_{i=0}^{n−1} ∑_{j=1}^{16} B_{j}^{i}.

Since sinh(x) and cosh(x) are odd and even functions, respectively, then

A_{1}^{i}(s)A_{5}^{i}(s) + A_{8}^{i}(s)A_{12}^{i}(s)
= 4C_{i}^{3}(s)S_{i}(s) − 2 cosh(\frac{Δ_{i}}{2} )C_{i}^{2}(s)S_{i}(s) + C_{i}^{3}(s)S_{i}(s) − cosh(\frac{Δ_{i}}{2} )C_{i}^{2}(s)S_{i}(s)

is anti-symmetric about s_{i}= \frac{s_{i}+s_{i+1}}{2} . Thus B_{2}^{i} = 0. Similarly, B_{j}^{i} = 0 for j = 3, 5, 8, 9, 12, 14, and 15..

For the remaining terms, we will use the fact that there exists a bound M such

that for all partitions P, and all (s, t) ∈ [0, 1]^{2}, |G_{P}(s, t)| < M and |(G_{P})_{s}(s, t)| < M (this can be verified by the definition of G_{P} and the continuity of sinh and cosh as in (?? )). We also use the fact that S_{i}(s) and C_{i}(s) are analytic everywhere with the following Taylor series about s =s_{i}:

S_{i}(s) = sinh(s− s_{i}) = ∑_{n=0}^{∞} \frac{(s− s_{i})^{2n+1}}{(2n + 1)!}
C_{i}(s) = cosh(s− s_{i}) = ∑_{n=0}^{∞} \frac{(s− s_{i})^{2n}}{(2n)!} .

For the B_{1}^{i} term we note that

A_{1}^{i}(s)A_{4}^{i}(s) + A_{8}^{i}(s)A_{11}^{i}(s)
= 4C_{i}^{2}(s)S_{i}^{2}(s) − 2 cosh(\frac{Δ_{i}}{2} )C_{i}(s)S_{i}^{2}(s) + C_{i}^{4}(s) − cosh(\frac{Δ_{i}}{2} )C_{i}^{3}(s).

By symmetry about s_{i}= \frac{s_{i}+s_{i+1}}{2} and the Taylor expansion of the above term, it is clear that

∫_{s_{i}}^{s_{i+1}}{A_{1}^{i}(s)A_{4}^{i}(s) + A_{8}^{i}(s)A_{11}^{i}(s)}ds = f(Δ_{i}),

where f(x) = o(x) and \frac{f(x)}{x} is increasing. Thus, by Lemma 3.2.2, there exists an ϵ_{4,2}(P) such that ϵ_{4,2}(P) → 0 as |P| → 0 and

∑_{i=0}^{n−1} B_{2}^{i} ≤ M ∑_{i=0}^{n−1} |K⟨h(s_{i}), h(s_{i})⟩ ∫_{s_{i}}^{s_{i+1}}{A_{1}^{i}(s)A_{4}^{i}(s) + A_{8}^{i}(s)A_{11}^{i}(s)}ds|
= M ∑_{i=0}^{n−1} |K⟨h(s_{i}), h(s_{i})⟩f(Δ_{i})|
≤ ϵ_{4,2}(P) ∥h∥^{2} .

For the B_{6}^{i} term we use Lemma 3.2.3 to get

∑_{i=0}^{n−1} B_{6}^{i} = ∑_{i=0}^{n−1} |K⟨Δ_{i}h, Δ_{i}h⟩G_{P}(s_{i}, s_{i})\frac{1}{η_{i}^{2}} ∫_{s_{i}}^{s_{i+1}}\{A_{2}^{i}(s)A_{5}^{i}(s) + A_{9}^{i}(s)A_{12}^{i}(s)\}ds|
≤ (∑_{i=0}^{n−1} |K⟨Δ_{i}h, Δ_{i}h⟩|)M sup_{i} \{|\frac{1}{Δ_{i}^{2}} ∫_{s_{i}}^{s_{i+1}}\{A_{2}^{i}(s)A_{5}^{i}(s) + A_{9}^{i}(s)A_{12}^{i}(s)\}ds|\}
≤ |P| ∥h∥^{2} M sup_{i} \{|\frac{1}{Δ_{i}^{2}} ∫_{s_{i}}^{s_{i+1}}\{A_{2}^{i}(s)A_{5}^{i}(s) + A_{9}^{i}(s)A_{12}^{i}(s)\}ds|\}

Considering the Taylor series expansion of S_{i}(s) and C_{i}(s), one can see that

∫_{s_{i}}^{s_{i+1}}{A_{2}^{i}(s)A_{5}^{i}(s) + A_{9}^{i}(s)A_{12}^{i}(s)}ds

= ∫_{s_{i}}^{s_{i+1}}\{2C_{i}^{2}(s)S_{i}^{2}(s) + 2C_{i}^{4}(s) − 2C_{i}^{3}(s) cosh(\frac{Δ_{i}}{2} )  (3.18)

+C_{i}^{2}(s)S_{i}^{2}(s) − C_{i}(s)S_{i}^{2}(s) cosh(\frac{Δ_{i}}{2} )\}ds

= O(Δ_{i}^{3})

where O(x) is a function with the property that there exist C > 0 and r > 0 such that \frac{|O(x)|}{|x|} < C for all |x| < r. It then follows that there exists an N > 0 such that for |P| sufficiently small,

sup_{i} \{|\frac{1}{Δ_{i}^{2}} ∫_{s_{i}}^{s_{i+1}}\{A_{2}^{i}(s)A_{5}^{i}(s) + A_{9}^{i}(s)A_{12}^{i}(s)\}ds|\} ≤ N.

Thus, ∑_{i=0}^{n−1} B_{6}^{i} ≤ ϵ_{4,6}(P) ∥h∥^{2}, where ϵ_{4,6}(P) = |P| MN.

For the B_{7}^{i} term we note that

A_{2}^{i}(s)A_{6}^{i}(s)+A_{9}^{i}(s)A_{13}^{i}(s) = 2S_{i}^{4}(s)−2C_{i}(s)S_{i}^{2}(s)+3C_{i}^{2}(s)S_{i}^{2}(s)−3C_{i}(s)S_{i}^{2}(s) cosh (\frac{Δ_{i}}{2} )

By symmetry about s_{i}= \frac{s_{i}+s_{i+1}}{2} and the Taylor expansion of the above term, it is clear that

\frac{1}{Δ_{i}} ∫_{s_{i}}^{s_{i+1}}\{A_{2}^{i}(s)A_{6}^{i}(s) + A_{9}^{i}(s)A_{13}^{i}(s)\}ds = g(Δ_{i}),

where g(x) = o(x) and \frac{g(x)}{x} is increasing. Next, by the Mean Value Theorem, for each i ∈ \{0, 1, …, n − 1\} there exists an ŝ_{i} ∈ (s_{i}, s_{i+1}) such that \frac{G_{P}(s_{i+1},s_{i})−G_{P}(s_{i},s_{i})}{Δ_{i}} = (G_{P})_{s}(ŝ_{i}, s_{i}). We then have

∑_{i=0}^{n−1} B_{7}^{i} = |∑_{i=0}^{n−1} K⟨Δ_{i}h, h(s_{i})⟩(G_{P}(s_{i+1}, s_{i}) − G_{P}(s_{i}, s_{i}))

·\frac{1}{η_{i}^{2}} ∫_{s_{i}}^{s_{i+1}} \{A_{2}^{i}(s)A_{6}^{i}(s) + A_{9}^{i}(s)A_{13}^{i}(s)\} ds|

≤ ∑_{i=0}^{n−1} |K⟨Δ_{i}h, h(s_{i})⟩(G_{P})_{s}(ŝ_{i}, s_{i})\frac{1}{Δ_{i}} ∫_{s_{i}}^{s_{i+1}} \{A_{2}^{i}(s)A_{6}^{i}(s) + A_{9}^{i}(s)A_{13}^{i}(s)\} ds|
≤ M ∑_{i=0}^{n−1} |K⟨Δ_{i}h, h(s_{i})⟩\frac{1}{Δ_{i}} ∫_{s_{i}}^{s_{i+1}} \{A_{2}^{i}(s)A_{6}^{i}(s) + A_{9}^{i}(s)A_{13}^{i}(s)\} ds| .

It thus follows by Lemma 3.2.2 that there exists an ϵ_{4,7}(P) such that ϵ_{4,7}(P) → 0 as

|P| → 0 and ∑_{i=0}^{n−1} B_{7}^{i} ≤ ϵ_{4,7}(P) ∥h∥^{2} .

For the B_{10}^{i} term, from (3.18) and the reasoning for the B_{7}^{i} term it follows that

there exists an ϵ_{4,10}(P) such that ϵ_{4,10}(P) → 0 as |P| → 0 and ∑_{i=0}^{n−1} B_{10}^{i} ≤ ϵ_{4,10}(P) ∥h∥^{2} .

For the B_{11}^{i} term, we note that the same reasoning for the B_{7}^{i} term applies to

show that there exists an ϵ_{4,11}(P) such that ϵ_{4,11}(P) → 0 as |P| → 0 and ∑_{i=0}^{n−1} B_{11}^{i} ≤ ϵ_{4,11}(P) ∥h∥^{2}.

Going back to the B_{4}^{i} term, we note that

A_{1}^{i}(s)A_{7}^{i}(s) + A_{8}^{i}(s)A_{13}^{i}(s) = 5C_{i}^{2}(s)S_{i}^{2}(s) − 3 cosh(\frac{Δ_{i}}{2} )C_{i}(s)S_{i}^{2}(s),

and thus the reasoning for the B_{7}^{i} term applies to show that there exists an ϵ_{4,4}(P) such

that ϵ_{4,4}(P) → 0 as |P| → 0 and ∑_{i=0}^{n−1} B_{4}^{i} ≤ ϵ_{4,4}(P) ∥h∥^{2}.

For the B_{13}^{i} term we note that

A_{3}^{i}(s)A_{4}^{i}(s) + A_{10}^{i}(s)A_{11}^{i}(s) = 5C_{i}^{2}(s)S_{i}^{2}(s) − \frac{1}{ς_{i}} η_{i}^{2}C_{i}(s)S_{i}^{2}(s) − \frac{1}{2ς_{i}} η_{i}^{2}C_{i}^{3}(s),

and so again the reasoning for the B_{7}^{i} term applies to show that there exists an ϵ_{4,13}(P)

such that ϵ_{4,13}(P) → 0 as |P| → 0 and ∑_{i=0}^{n−1} B_{13}^{i} ≤ ϵ_{4,13}(P) ∥h∥^{2} .

Finally, for the B_{16}^{i} term we note that, as in (3.18),

∫_{s_{i}}^{s_{i+1}}{A_{3}^{i}(s)A_{7}^{i}(s) + A_{10}^{i}(s)A_{14}^{i}(s)}ds
= ∫_{s_{i}}^{s_{i+1}}\{4C_{i}^{2}(s)S_{i}^{2}(s) + S_{i}^{4}(s) − \frac{3}{ς_{i}} η_{i}^{2}C_{i}(s)S_{i}^{2}(s)\}ds
= O(Δ_{i}^{3}).

So there exists an N > 0 such that for |P| sufficiently small,

sup_{i} |\frac{1}{Δ_{i}^{3}} ∫_{s_{i}}^{s_{i+1}}\{A_{3}^{i}(s)A_{7}^{i}(s) + A_{10}^{i}(s)A_{14}^{i}(s)\}ds| ≤ N.

Applying the Mean Value Theorem to G_{P}(·, s_{i+1}) and G_{P}(·, s_{i}) and using Lemma 3.2.3,

we get ∑_{i=0}^{n−1} B_{16}^{i} ≤ ϵ_{4,16}(P) ∥h∥^{2}, where ϵ_{4,16}(P) = 2MN |P| .

We have thus shown that

| ∑_{k∈S_{P}}(2∇_{k}(Q_{P}[h, k]), h)| ≤ ϵ_{4}(P) ∥h∥^{2} ,

where

ϵ_{4}(P) ≡ ϵ_{4,1}(P) + ϵ_{4,4}(P)ϵ_{4,6}(P) + ϵ_{4,7}(P)
+ϵ_{4,10}(P) + ϵ_{4,11}(P) + ϵ_{4,13}(P) + ϵ_{4,16}(P).

The proof of Theorem 3.2.1 is now complete by noting that

|((Ric − Ric_{P})h, h)| ≤ ϵ(P) ∥h∥^{2}

where ϵ(P) = ϵ_{1}(P) + ϵ_{2}(P) + ϵ_{3}(P) + ϵ_{4}(P).□

The Heat Kernel Measure

For each partition, P, of [0, 1], let B_{P} be the Borel σ-algebra on G^{P}. We define

G to be the smallest σ-algebra on L(G) such that the projections π_{P} : L(G) → G^{P} are G/B_{P}-measurable for each partition, P. Our goal in this section is to show that for each fixed g_{0} ∈ L(G) and each t > 0, there exists a unique heat kernel probability measure ν_{t}(g_{0}, ·) on (L(G), G), i.e., with the property that

(e^{t \frac{Δ}{2}} f)(g_{0}) = ∫_{L(G)} f(g)ν_{t}(g_{0}, dg)

for all bounded f ∈ FC^{∞}(L(G)), where Δ is defined below in (4.3) and FC^{∞}(L(G)) are the cylinder functions defined in Section 1.2. We define (e^{t \frac{Δ}{2}} f)(g_{0}) to be the unique function, u(t, g_{0}), on [0, ∞) × L(G) that solves the heat equation:

\frac{∂u}{∂t}(t, g_{0}) = \frac{1}{2}Δ_{g_{0}}u(t, g_{0})
lim_{t↘0} u(t, g_{0}) = f(g_{0}).

We note that \{P_{t} ≡ e^{t \frac{Δ}{2}} \}_{t≥0} is the semigroup of operators on FC^{∞}(L(G)) with generator L ≡ \frac{Δ}{2} (i.e., Lf = lim_{t↘0} \frac{P_{t}f−f}{t} ∀ f ∈ FC^{∞}(L(G))).

The heat kernel measure will be derived from an L(G)-valued Brownian motion.

The existence of such a process is due to Malliavin [17]. The procedure needed to construct the process for the free loop group is virtually the same as that for the pinned loop group as described by Driver [3], so we will only summarize it and verify any steps that would be different in the free loop case.

The procedure begins by constructing an L(𝔤)-valued Brownian motion, { β_{t}}_{t≥0},

where L(𝔤) ≡ {k ∈ C([0, 1], 𝔤) : k(0) = k(1)}. We must first say what we mean by a “Brownian motion” on this space. Let

H^{BV} ≡ {h ∈ H(𝔤) : ∃ rt. cts. function, λ_{h}, of bounded
variation with h′(s) = λ_{h}(s)a.e.} .

For each h ∈ H^{BV}, define α_{h} ∈ L(𝔤)^{∗} by

α_{h}(k) ≡ − ∫_{0}^{1}⟨k(s), dλ_{h}(s)⟩ + ∫_{0}^{1}⟨k(s), h(s)⟩ds ∀ k ∈ L( 𝔤),  (4.1)

where the first integral is defined in the Lebesgue-Stieljes sense. Note that for h, k ∈ H^{BV} ⊆ H(𝔤), α_{h}(k) = (h, k)_{H(𝔤)}.

For a given filtered probability space (W, {F_{t}}_{t≥0}, F, P) satisfying the usual

conditions (i.e., complete and right-continuous), we say that {β_{t}}_{t≥0} is an L(𝔤)-valued Brownian motion on W provided

1. β_{t} is F_{t}/G-measurable ∀ t > 0,

2. for all ω ∈ W, the map t ↦ β_{t}(ω) is continuous from [0, ∞) to L(𝔤), and

3. {β_{t}}_{t≥0} is a mean-zero Gaussian process with covariance

E[α_{h}(β_{t})α_{k}(β_{s})] = t ∧ τ(h, k)

for all h, k ∈ H^{BV} and t, τ ∈ [0, ∞).

It is a well know fact that such a Brownian motion exists so we will not prove

it here (one method of proof uses Kolmogorov’s extension theorem and the observation that (H(𝔤), L(𝔤)) is an abstract Wiener space.

We will next show that an L(G)-valued Brownian motion on W may be derived

from {β_{t}}_{t≥0}. First, let us recall some definitions from the theory of stochastic integration. In general, if H and K are Hilbert spaces, let B(H, K) be the space of bounded linear mappings from H → K. If M and X are H and B(H, K)-valued semimartingales, respectively, then the following are the forward, backwards, and Stratonovich stochastic integrals, respectively, of X with respect to M:

∫_{0}^{t} XdM ≡ lim_{|π|→0} ∑_{i=0}^{∞} X_{t_{i}^{π}}(M_{t∧t_{i+1}^{π}} − M_{t∧t_{i}^{π}}),
∫_{0}^{t} X\overleftarrow{dM} ≡ lim_{|π|→0} ∑_{i=0}^{∞} X_{t_{i+1}^{π}}(M_{t∧t_{i+1}^{π}} − M_{t∧t_{i}^{π}}),
∫_{0}^{t} XδM ≡ lim_{|π|→0} ∑_{i=0}^{∞} \frac{X_{t_{i}^{π}} + X_{t_{i+1}^{π}}}{2} (M_{t∧t_{i+1}^{π}} − M_{t∧t_{i}^{π}}),

where the limits are in probability and each π = {0 = t_{0}^{π} < t_{1}^{π} < t_{2}^{π} < ⋯ ∞} is a partition of [0, ∞) .

The next theorem is due to Malliavin [17]. Assume 𝔤_{0} is an orthonormal basis

of 𝔤 = Lie(G), and (W, {F_{t}}_{t≥0}, F, P) is the filtered probability space from above.

Theorem 4.0.1 Suppose that G is a compact Lie group with Ad_{G} -invariant inner product, ⟨·, ·⟩, on 𝔤. Then for each g_{0} ∈ L(G), there exists a continuous adapted L(G) -valued process, {Σ_{t}}_{t≤0}, on W which solves the following family of Stratonovich differential equations:

Σ_{δt}(s) = ∑_{A∈𝔤_{0}} Ã(Σ_{t}(s))β_{δt}^{A}(s)
Σ_{0}(s) = g_{0}(s) ∀ s ∈ [0, 1],  (4.2)

where Ã is the left-invariant vector field on G with Ã(e) = A, and β_{t}^{A}(s) ≡ ⟨A, β_{t}(s)⟩.

The interpretation of (4.2), as defined in Chapter IV of Ikeda and Watanabe

[12], is that for any f ∈ C^{∞}(G),

f(Σ_{t}(s)) − f(Σ_{0}(s)) = ∑_{A∈𝔤_{0}} ∫_{0}^{t} Ãf(Σ_{τ} (s))β_{δτ}^{A} (s)

for all s ∈ [0, 1]. The existence of a solution to (4.2) is actually proved in Ikeda and Watanabe [12]. It is the existence of a jointly continuous version of the solution that requires further work. The proof supplied by Driver [3] for the existence of such a jointly continuous version in the pinned loop case can be applied verbatim for the free loop case. The key to the proof is the application of Kolmogorov’s continuity criterion. The only difference in our case is that we are using a different Green’s function, G(s, t), as defined in (2.4). We only need to note that G(s, t) and G_{s}(s, t) are uniformly bounded for all (s, t) ∈ [0, 1]^{2}, then the argument in [3] goes through.

We now explain why {Σ_{t}}_{t≥0} can be called a “Brownian motion” on L(G).

First note that the Laplacian, Δ, on L(G) is defined by

Δf ≡ ∑_{h∈S} ∇_{h}^{2}f ≡ ∑_{h∈S} ∇_{h}(∇_{h}f),  (4.3)

for all f ∈ FC^{∞}(L(G)), where S is an orthonormal basis for H(𝔤). We define ∇_{h}f by

(∇_{h}f)(g) ≡ h̃f(g)
= \frac{d}{dt} |_{t=0} f(ge^{th}) ∀ g ∈ L(G),

where t ⟼ ge^{th} is a path in L(G) with

\frac{d}{dt} |_{t=0} ((ge^{th})(s)) = h̃(s) |_{g(s)}∈ T_{g(s)}G

for each s ∈ [0, 1]. Note that if f = F ∘ π_{P} with, P = {0 = s_{0} < s_{1} < ⋯ < s_{n−1} < 1}, then

(∇_{h}f)(g) = ∑_{i=0}^{n−1}(h(s_{i})^{(i)}F)(π_{P}g),

where for any A ∈ 𝔤, A^{(i)} is defined to be the left invariant vector field on G^{P} given by

(A^{(i)}F)(g_{0}, …, g_{n−1}) ≡ \frac{d}{dr} |_{r=0} F (g_{0}, …, g_{i−1}, g_{i}e^{rA}, g_{n+1}, …, g_{n−1}), (4.4)

where e is the exponential map from 𝔤 to G (see Section 3.2 in do Carmo [7]).

Theorem 4.0.2 Suppose P = {0 = s_{0} < s_{1} < ⋯ < s_{n−1} < 1} is a given partition and T > 0. Suppose f : [0, T] × L(G) → R is a function of the form f(t, g) = F (t, g_{P}), where g_{P} = π_{P}g ∈ G^{P}, such that F : [0, T] × G^{P} → R is continuous, F |_{(0,T)×G^{P}} is smooth with first and second order derivatives extending to continuous functions on [0, T] × G^{P}. Then {M_{t}}_{t∈[0,T]} , defined by

M_{t} ≡ f(t, Σ_{t}) − f(0, Σ_{0}) − ∫_{0}^{t}((\frac{∂}{∂t} + \frac{1}{2}Δ)f(τ, ·))(Σ_{τ})dτ ∀ t ∈ [0, T],  (4.5)

is a martingale. In analogy with the finite dimensional case, we take this to be what defines {Σ_{t}}_{t≥0} as a Brownian motion.

Proof: Let (Σ_{P})_{t} ≡ (Σ_{t}(s_{0}), …, Σ_{t}(s_{n−1})) ∈ G^{P} for all t ∈ [0, 1]. By the fact

that Σ_{t}(s_{i}) solves Equation (4.2) for each s_{i} ∈ P, it can be shown that (Σ_{P})_{t} satisfies

δ(Σ_{P})_{t} = ∑_{i=0}^{n−1} ∑_{A∈𝔤_{0}} A^{(i)}((Σ_{P})_{t})β_{δt}^{A}(s_{i}).

By an application of Ito’s formula, we get

d[f(t, Σ_{t}) = \frac{∂}{∂τ} f(τ, Σ_{t}) |_{τ=t} + ∑_{i=0}^{n−1} ∑_{A∈𝔤_{0}}(A^{(i)}F (t, ·))((Σ_{P})_{t})β_{dt}^{A}(s_{i}) + \frac{1}{2}Δf(t, Σ_{t})dt.

Thus,

M_{t} = ∑_{i=0}^{n−1} ∑_{A∈𝔤_{0}} ∫_{0}^{t}(A^{(i)}F (τ, ·))((Σ_{P})_{τ})β_{δτ}^{A} (s_{i}).

Since {β_{t}^{A}(s_{i})}_{t≥0} is a martingale for each s_{i} and (A^{(i)}F (τ, ·))((Σ_{P})_{τ}) is bounded

for each i and A, then it is well known that this implies {M_{t}}_{t≥0} is a martingale (see Theorem 20 on p. 20 in Protter [18]). □

We are now ready to define the heat kernel measure on (L(G), G). For each

t > 0, let ν_{t}(e, ·) be the law of Σ_{t}, where e ∈ L(G) is the identity loop. In other words,

ν_{t}(e, A) ≡ P (Σ_{t} ∈ A) ∀ ∈ G.

And for any g_{0} ∈ L(G), define ν_{t} (g_{0}, ·) by

ν_{t}(g_{0}, A) = (L_{g_{0}∗}ν_{t}(e, ·))(A)
≡ ν_{t}(e, L_{g_{0}}^{−1}(A))
= P (L_{g_{0}} ∘ Σ_{t} ∈ A).  (4.6)

Theorem 4.0.3 For all bounded f ∈ FC^{∞}(L(G)),

(e^{t \frac{Δ}{2}} f)(g_{0}) = ∫_{L(G)} f(g)ν_{t}(g_{0}, dg) ∀ g_{0} ∈ L(G).

Proof: Let f ∈ FC^{∞}(L(G)) be of the form f(g) = F (g_{P}), F ∈ C^{∞}(G^{P}), for a

given partition P, and let g_{0} ∈ L(G). Define

u(t, g_{0}) ≡ ∫_{L(G)} f(g)ν_{t}(g_{0}, dg) ∀ t ≥ 0.

Then by (4.6),

u(t, g_{0}) = E[f(L_{g_{0}} ∘ Σ_{t})] = E[f̃(g_{0}, Σ_{t})] = E[F̃(g_{0}, (Σ_{P})_{t})],

where f̃(g_{0}, ·) ≡ f ∘ L_{g_{0}}(·), F̃(g_{0}, ·) ≡ F ∘ L_{(π_{P}∘g_{0})}(·) ∈ C^{∞}(G^{P}), and (Σ_{P})_{t} ≡ π_{P}(Σ_{t}). From (4.5) it follows that

u(t, g_{0}) = E[M_{t} + f̃(g_{0}, Σ_{0}) + ∫_{0}^{t} \frac{1}{2}Δ_{2}f̃(g_{0}, Σ_{τ})dτ]
= E[f̃(g_{0}, Σ_{0})] + ∫_{0}^{t} E(\frac{1}{2}Δ_{2}f̃(g_{0}, Σ_{τ}))dτ,

where we have used the fact that M_{t} is a martingale with M_{0} = 0 a.e.. We will use the notation Δ_{2} and Δ_{P,2}, where appropriate, to indicate that we are applying Δ or Δ_{P} to the second variable of the function on which they are operating. For example, Δ_{2}f̃(g_{0}, Σ_{τ}) ≡ Δ(f̃(g_{0}, ·))(Σ_{τ}). It is easy to check that Δ_{2}f̃(g_{0}, Σ_{τ}) = Δ_{P,2}F̃(g_{0}, (Σ_{P})_{τ}) , where

Δ_{P,2}F̃(g_{0}, g_{P}) = ∑_{A∈𝔤_{0}} ∑_{i,j=0}^{n−1} G(s_{i}, s_{j})(A^{(i)}A^{(j)}F̃(g_{0}, ·))(g_{P})

for all g_{P} ∈ G^{P}, and

(Σ_{P})_{τ} ≡ π_{P}(Σ_{τ}) = (Σ_{τ} (s_{0}), …, Σ_{τ} (s_{n−1})) ∈ G^{P}.

A^{(i)} is the notation we introduced in (4.4). We thus have

u(t, g_{0}) = E[F̃(g_{0}, (Σ_{P})_{0})] + ∫_{0}^{t} E[\frac{1}{2}Δ_{P,2}F̃(g_{0}, (Σ_{P})_{τ})]dτ.

Intuitively, we would like for E[\frac{1}{2}Δ_{P,2}F̃(g_{0}, (Σ_{P})_{τ})] = \frac{1}{2}Δ_{P,g_{0}}E[F̃(g_{0}, (Σ_{P})_{τ})].

This is indeed a true statement, but not trivial. One may refer to the chapter on diffusion processes on manifolds (Chapter V, p. 254) in Ikeda and Watanabe [12] for a proof. We can then write

u(t, g_{0}) = F̃(g_{0}, (Σ_{P})_{0}) + ∫_{0}^{t} \frac{1}{2}Δ_{P,g_{0}}E[F̃(g_{0}, (Σ_{P})_{τ})]dτ,

and thus

\frac{∂u}{∂t}(t, g_{0}) = \frac{1}{2}Δ_{P,g_{0}}E[F̃(g_{0}, (Σ_{P})_{t})] = \frac{1}{2}Δ_{g_{0}}E[f̃(g_{0}, Σ_{t})]
= \frac{1}{2}Δ_{g_{0}}u(t, g_{0}).

And by the dominated convergence theorem and the fact that Σ_{0} = e a.e., we have

lim_{t↘0} u(t, g_{0}) = lim_{t↘0} E[f(L_{g_{0}} ∘ Σ_{t})]
= E[f(L_{g_{0}} ∘ e)]
= E[f(g_{0})] = f(g_{0}).

Therefore, we have

(e^{t \frac{Δ}{2}} f)(g_{0}) = ∫_{L(G)} f(g)ν_{t}(g_{0}, dg). □

For a fixed g_{0P} ∈ G^{P} and t > 0, define the probability measure ν_{t}^{P}(g_{0P}, ·) on

G^{P} by

ν_{t}^{P}(g_{0P}, A) ≡ P (L_{g_{0P}} ∘ (Σ_{P})_{t} ∈ A) ∀ A ∈ G^{P}.

It then follows from the above discussion that {(Σ_{P})_{t}}_{t≥0} is a diffusion process on G^{P} with generator Δ_{P} and that

(e^{t \frac{Δ_{P}}{2}} F)(g_{0P}) = ∫_{G^{P}} F (g_{P})ν_{t}^{P}(g_{0P}, dg_{P})  (4.7)

for all F ∈ C^{∞}(G^{P}) and g_{0P} ∈ G^{P}.

The following lemma will be useful in proving the logarithmic Sobolev inequal-

ity.

Lemma 4.0.4 For a partition P = {0 = s_{0} < s_{1} < ⋯ < s_{n−1} < 1},

Δ_{P} = ∑_{h∈Γ} h̃^{2},

where Γ is an orthonormal basis of (𝔤^{P}, (·, ·)_{P}) and (·, ·)_{P} is defined as in (3.1).

Proof: For each s_{i} ∈ P, let G_{P}(s_{i}) ≡ (G_{P}(s_{i}, s_{0}), …, G_{P}(s_{i}, s_{n−1})) ∈ R^{n}, and

for A ∈ 𝔤, G_{P}(s_{i})A ≡ (G_{P}(s_{i}, s_{0})A, …, G_{P}(s_{i}, s_{n−1})A) ∈ 𝔤^{P}. For F ∈ C^{∞}(G^{P}) and g_{P} ∈ G^{P} , we then have

∑_{h∈Γ} h̃^{2}F (g_{P}) = ∑_{h∈Γ} h̃( ∑_{A∈𝔤_{0}} ∑_{i=0}^{n−1}(A^{(i)}F (g_{P}))⟨A, h_{i}⟩
= ∑_{h∈Γ} ∑_{A,B∈𝔤_{0}} ∑_{i,j=0}^{n−1}(B^{(j)}A^{(i)}F (g_{P}))⟨A, h_{i}⟩⟨B, h_{j}⟩,

where each h ∈ Γ is denoted by h = (h_{0}, h_{1}, …, h_{n−1}). Recalling that (Q_{ij}) = (G_{ij})^{−1},

⟨A, h_{i}⟩ = ∑_{k,l=0}^{n−1} Q_{kl}⟨G(s_{i}, s_{k})A, h_{l}⟩
= (G_{P}(s_{i})A, h)_{P}.

Using the same calculations for ⟨B, h_{j}⟩, we see that

∑_{h∈Γ} h̃^{2}F (g_{P}) = ∑_{h∈Γ} ∑_{A,B∈𝔤_{0}} ∑_{i,j=0}^{n−1}(B^{(j)}A^{(i)}F (g_{P}))(G_{P}(s_{i})A, h)_{P}(G_{P}(s_{j})B, h)_{P}
= ∑_{A,B∈𝔤_{0}} ∑_{i,j=0}^{n−1}(B^{(j)}A^{(i)}F (g_{P}))(G_{P}(s_{i})A, G_{P}(s_{j})B)_{P}
= ∑_{A,B∈𝔤_{0}} ∑_{i,j=0}^{n−1}(B^{(j)}A^{(i)}F (g_{P})){ ∑_{k,l=0}^{n−1} Q_{kl}G(s_{i}, s_{k})G(s_{j}, s_{l})⟨A, B⟩}
= ∑_{A∈𝔤_{0}} ∑_{i,j=0}^{n−1}(A^{(j)}A^{(i)}F (g_{P}))G(s_{i}, s_{j}) = Δ_{P}F (g_{P}). □

We next wish to observe that since (·, ·)_{P} is left invariant on G^{P}, then the

Riemannian volume measure, λ_{P}, on G^{P} is a left Haar measure (i.e., λ_{P} is a left invariant, nonzero Radon measure). And since G^{P} is compact, it is thus unimodular, i.e., λ_{P} is both a left and right Haar measure (see Proposition 10.16 in Folland [8]]). It is well known that in this case, ∑_{h∈Γ} h̃^{2} , and thus Δ_{P}, is the Laplace Beltrami operator on G^{P} (see Proposition 5.1 in Driver [3] and Remark (2.2) in Driver and Gross [6]. Using this fact, it then follows by Theorem 2.1 in Driver [3] that for t > 0, there exists a smooth function p_{t}(x_{P}, g_{P}) on G^{P} × G^{P} such that

(e^{t \frac{Δ_{P}}{2}} F)(x_{P}) = ∫_{G^{P}} p_{t}(x_{P}, g_{P})F (g_{P})λ_{P}(dg_{P}) ∀ F ∈ C^{∞}(G^{P}).

By (4.7), we then have

∫_{G^{P}} F (g_{P})ν_{t}^{P}(g_{0P}, dg_{P}) = ∫_{G^{P}} F (g_{P})p_{t}(g_{0P}, g_{P})λ_{P}(dg_{P})

for all g_{0P} ∈ G^{P} and F ∈ C^{∞}(G^{P}).

Integration By Parts on L(G)

The motivation for developing an integration by parts formula on L(G) is re-

lated to the task of making our logarithmic Sobolev inequality a general as possible. In other words, we want the inequality to hold for a large class of functions. This will be accomplished by showing the closability of a certain symmetric, quadratic form that appears in the inequality.

5.1 Closed and Closable Quadratic Forms

Let us take a moment to review some facts about quadratic forms. One may

refer to Ma and Röckner [16] or Fukushima, Oshima, and Takeda [9] for more on this subject. Let H be a general Hilbert space (H is not necessarily the same as H(𝔤) from above). Let E : D(E) × D(E) → R be a bilinear form on a linear subspace D(E) → H. We will assume D(E) is dense in H with respect to the Hilbert norm on H, and E is positive symmetric. Let E : D(E) → R also denote the associated quadratic form defined by

E(v) ≡ E(v, v) ∀ v ∈ D(E).

Define the inner product, (·, ·)_{1}, on D(E) by

(v, w)_{1} ≡ (v, w)_{H} + E(v, w) ∀ v, w ∈ D(E),

where (·, ·)_{H} is the Hilbert inner product on H. Define ∥·∥_{1} by ∥v∥_{1} ≡ \sqrt{(v, v)_{1}} for all v ∈ D(E).

Definition 5.1.1 E is closed if and only if (D(E), (·, ·)_{1}) is a Hilbert space.

The following proposition is easy to verify.

Proposition 5.1.2 E is closed if and only if for all sequences {v_{n}} ⊆ D(E) such that ∥v_{n} − v∥_{H} → 0 as n → ∞ for some v ∈ H and E(v_{n} − v_{m}) → 0 as m, n → ∞, we have v ∈ D(E) and E(v_{n} − v) → 0 as m, n → ∞.

If (D(E), (·, ·)_{1}) is not a Hilbert space, it may be possible to extend E to a closed

quadratic form, \overline{E}, on some larger space H_{1} ⊃ D(E). This is possible when E is closable.

Definition 5.1.3 E is closable if and only if for all sequences {v_{n}} ⊆ D(E) such that ∥v_{n}∥_{H} → 0 as n → ∞ and E(v_{n} − v_{m}) → 0 as m, n → ∞, we have E(v_{n}) → 0 as n → ∞.

If E is closable, we can define H_{1} to be H_{1} ≡ C/ ∼, where C is the set of Cauchy

sequences in (D(E), (·, ·)_{1}), i.e.,

C ≡ {{v_{n}} ⊆ D(E) : lim_{m,n→∞} ∥v_{n} − v_{m}∥_{1} = 0},

and ∼ is the equivalence relation

{v_{n}} ∼ {w_{n}} ⇔ lim_{n→∞} ∥v_{n} − w_{n}∥_{1} = 0.

H_{1} is thus the set of all equivalence classes in C.

Given v̄, w̄ ∈ H_{1}, where v̄ and w̄ are the equivalence classes of {v_{n}} and {w_{n}},

define ((·, ·))_{1} on H_{1} by

((v̄, w̄))_{1} ≡ lim_{n→∞}(v_{n}, w_{n})_{1}.

One can check that the definition of ∼ implies that ((·, ·))_{1} is well-defined.

We can see that D(E) is imbedded in H_{1} by the map

ι : D(E) → H_{1}
v ↦ v̄,

where v̄ is the equivalence class of {v_{n}} and v_{n} = v ∀ n ≥ 1. Clearly, ι is an isometry. We can also imbed H_{1} in H by

ι̂: H_{1} → H

v̄ ↦ lim_{n→∞} v_{n},

where {v_{n}} is a representative of the equivalence class of v̄ and the limit is with respect to ∥·∥_{H}. Since {v_{n}} is Cauchy in ∥·∥_{1} and ∥·∥_{H} ≤ ∥·∥_{1}, then {v_{n}} is Cauchy in ∥·∥_{H} and thus lim_{n→∞} v_{n} is guaranteed to exist.

Note that ι̂ is injective precisely due to the fact that E is closable. Indeed, if

{v_{n}} ∈ v̄ ∈ H_{1} and {w_{n}} ∈ w̄ ∈ H_{1} with lim_{n→∞} v_{n} = lim_{n→∞} w_{n}, then ∥v_{n} − w_{n}∥_{H} → 0 as n → ∞ and

E((v_{n} − w_{n}) − (v_{m} − w_{m})) ≤ E(v_{n} − v_{m}) − E(w_{n} − w_{m}) → 0 as m, n → ∞.

Hence, the closability of E gives E(v_{n} − w_{n}) → 0 as n → ∞ and thus ∥v_{n} − w_{n}∥_{1} → 0 as n → ∞.

The form E can be extended to H_{1} by defining \overline{E} : H_{1} × H_{1} → C by
\overline{E}(v̄, w̄) ≡ lim_{n→∞} E(v_{n}, w_{n}) ∀ v̄, w̄ ∈ H_{1},  (5.1)

where {v_{n}} ∈ v̄ and {w_{n}} ∈ w̄. (One may use the Cauchy-Schwartz inequality and the fact that {v_{n}} is ∥·∥_{1}-Cauchy to show the limit on the right in (5.1) exists.)

If we identify H_{1} with ι̂(H_{1}), i.e., let

H_{1} ≡ {v ∈ H : ∃ a (·, ·)_{1} -Cauchy sequence {v_{n}} ⊆ D(E)
with ∥v_{n} − v∥_{H} → 0 as n → ∞} ,

then we can extend E to \overline{E} on H_{1} ⊆ H in the same way as above, i.e.,

E(v, w) ≡ lim_{n→∞} E(v_{n}, w_{n}) ∀ v, w ∈ H_{1},

where {v_{n}}, {w_{n}} ⊆ C and ∥v_{n} − v∥_{H} → 0, ∥w_{n} − w∥_{H} → 0 as n → ∞. This allows us to view H_{1} as an actual subspace of H and not a set of equivalence classes.

Now let H and K both be Hilbert spaces. It will be useful for us to consider the

closability of a particular type of form defined in terms of a linear operator A : H → K as follows:

E_{A} : D(E_{A}) × D(E_{A}) → C
E_{A}(v, w) ≡ (Av, Aw)_{K} ∀ v, w ∈ D(A),

where D(E_{A}) = D(A), the domain of A, is assumed to be dense in H. As we will show below, it turns out that E_{A} is closable if and only if A is a closable operator.

Letting L(H, K) denote the space of linear operators from H to K, and Γ(A)

denote the graph of A ∈ L(H, K), i.e.,

Γ(A) ≡ {(v, Av) : v ∈ D(A)} ⊆ H × K,

recall the following definitions.

Definition 5.1.4 A ∈ L(H, K) is closed iff Γ(A) is closed in H × K. A ∈ L(H, K) is closable iff \overline{Γ(A)} is the graph of an operator in L(H, K).

If A is closable, we will let \overline{A} be its closure, i.e., \overline{A} is the linear operator with

the property Γ(\overline{A}) = \overline{Γ(A)}. It is not hard to check that A is closable if and only if given {v_{n}} ⊆ D(A) with∥v_{n}∥_{H} → 0 and ∥Av_{n} − u∥_{K} → 0 as n → ∞ for some u ∈ K, then u = 0. The next lemma gives a very convenient necessary and sufficient condition for closability of a linear operator which will be used in Theorem 5.6.1.

Lemma 5.1.5 A ∈ L(H, K) is closable if and only if D(A^{∗}) is dense in K.

Proof: Recall that A^{∗} : D(A^{∗}) ⊆ K → H is the adjoint of A where u ∈ D(A^{∗})

if and only if ∃ w ∈ H such that

⟨Av, u⟩ = ⟨v, w⟩ ∀ v ∈ D(A),

in which case A^{∗}u ≡ w.

We first introduce the following notation. Let (·, ·)_{K×H} be the inner product

on K × H defined by

⟨(u_{1}, v_{1}), (u_{2}, v_{2})⟩_{K×H} ≡ ⟨u_{1}, u_{2}⟩_{K} + ⟨v_{1}, v_{2}⟩_{H} ∀ (u_{1}, v_{1}), (u_{2}, v_{2}) ∈ K × H.

Let T : H × K → K × H be the unitary operator

T : (v, u) ⟼ (u, −v).

Note that Γ(A^{∗}) = [T Γ(A)]^{⊥} by the following:

(u, v) ∈ Γ(A^{∗}) ⇔ ⟨Aw, u⟩_{K} = ⟨w, v⟩_{H} ∀ w ∈ D(A)
⇔ ⟨Aw, u⟩_{K} + ⟨−w, v⟩_{H} = 0 ∀ w ∈ D(A)
⇔ ⟨(Aw, −w), (u, v)⟩_{K×H} = 0 ∀ w ∈ D(A)
⇔ ⟨T (w, Aw), (u, v)⟩_{K×H} = 0 ∀ w ∈ D(A)
⇔ (u, v)⊥T Γ(A).

It is a well-known fact that since \overline{Γ(A)} is closed, then \overline{Γ(A)} = (\overline{Γ(A)}^{⊥})^{⊥} (see

Lemma 3.3-6 in Kreyszig [14]). But by the continuity of the inner product on H × K, \overline{Γ(A)}^{⊥} = Γ(A)^{⊥}, so \overline{Γ(A)} = (Γ(A)^{⊥})^{⊥}. The fact that T is unitary then gives

\overline{Γ(A)} = (Γ(A)^{⊥})^{⊥}
= T^{−1}((T Γ(A)^{⊥})^{⊥})
= T^{−1}(Γ(A^{∗})^{⊥}).

We can now finish the proof by contradiction:

A is not closable ⇔ ∃ u ≠ 0 in K and {v_{n}} ⊆ D(A) with
v_{n} → 0 and Av_{n} → u as n → ∞
⇔ (0, u) ∈ \overline{Γ(A)} = T^{−1}(Γ(A^{∗})^{⊥})
⇔ T (0, u) = (u, 0) ∈ Γ(A^{∗})^{⊥}
⇔ ⟨(u, 0), (w, A^{∗}w)⟩_{K×H} = 0 ∀ w ∈ D(A^{∗})
⇔ ⟨u, w⟩_{K} = 0 ∀ w ∈ D(A^{∗})
⇔ u⊥D(A^{∗}) for some u ≠ 0 in K
⇔ D(A^{∗})^{⊥} ≠ {0}
⇔ K\\overline{D(A^{∗})} ≠ ϕ.

See Lemma 3.3-7 in Kreyszig [14] for an explanation of the last equivalence above. □ We can now make the following link between A and E_{A}.

Lemma 5.1.6 E_{A} is closable if and only if A is closable.

Proof: Suppose E_{A} is closable. By definition, this means that if {v_{n}} ⊆ D(E_{A})

with ∥v_{n}∥_{H} → 0 and E_{A}(v_{n} − v_{m}) → 0 as m, n → ∞, then E_{A}(v_{n}, v_{n}) = ∥Av_{n}∥_{K} → 0. To show A is closable, we assume {v_{n}} ⊆ D(A) = D(E_{A}) with ∥v_{n}∥_{H} → 0 and ∥Av_{n} − u∥_{K} → 0 for some u ∈ K and show this implies u = 0. But ∥Av_{n} − u∥_{K} → 0 implies {Av_{n}} is Cauchy in K and thus E_{A}(v_{n}−v_{m}) = ∥Av_{n} − Av_{m}∥_{K} → 0 as m, n → ∞. By the closability of E_{A}, we see that ∥Av_{n}∥_{K} = E_{A}(v_{n}, v_{n}) → 0 as n → ∞. By uniqueness of limits, it must be that u = 0.

Conversely, suppose A is closable and {v_{n}} ⊆ D(E_{A}) = D(A) with ∥v_{n}∥_{H} → 0

and E_{A}(v_{n} − v_{m}) → 0 as m, n → ∞. Then ∥Av_{n} − Av_{m}∥_{K} → 0 as m, n → ∞, and since K is complete, ∃ u ∈ K with ∥Av_{n} − u∥_{K} → 0. Since A is closable, then u = 0 and thus E_{A}(v_{n}, v_{n}) = ∥Av_{n}∥_{K} → 0, proving that E_{A} is closable. □

The quadratic form appearing in the logarithmic Sobolev inequality will be

denoted by E_{g_{0},t} : D(E_{g_{0},t}) × D(E_{g_{0},t}) → R, for each t ≥ 0 and g_{0} ∈ L(G), where

E_{g_{0},t}(f, g) ≡ ∫_{L(G)}(∇f(x), ∇g(x))_{H(𝔤)}ν_{t}(g_{0}, dx),
E_{g_{0},t}(f) ≡ ∫_{L(G)} ∥∇f(x)∥_{H(𝔤)}^{2} ν_{t}(g_{0}, dx),

and D(E_{g_{0},t}) ≡ FC^{∞}(L(G)). E_{g_{0},t} is clearly symmetric and positive. By definition, if f ∈ FC^{∞}(L(G)) is of the form f(g) = F (g_{P}) ,where F ∈ C^{∞}(G^{P}), and if g ∈ L(G), we define ∇f(g) to be the unique element in H(𝔤) such that

(∇f(g), h) = (h̃f)(g) ∀ h ∈ H(𝔤).  (5.2)

In our case, for all h ∈ H(𝔤), the fact that our Green’s function, G(s, t), is a reproducing kernel gives

( ∑_{A∈𝔤_{0}} ∑_{i=0}^{n−1}(A^{(i)}F (t, ·))(g_{P})G(s_{i}, ·)A, h) = ∑_{A∈𝔤_{0}} ∑_{i=0}^{n−1}(A^{(i)}F (t, ·))(g_{P})⟨A, h(s_{i})⟩
= ∑_{i=0}^{n−1}(h(s_{i})^{(i)}F (t, ·))(g_{P}) = (h̃f(t, ·))(g),

where h(s_{i})^{(i)} is defined as in (4.4). Thus,

(∇f(t, ·))(g) = ∑_{A∈𝔤_{0}} ∑_{i=0}^{n−1}(A^{(i)}F (t, ·))(g_{P})G(s_{i}, ·)A.  (5.3)

The Hilbert space in our case is L^{2}(L(G), ν_{t}(g_{0}, ·)) ⊃ D(E_{g_{0},t}). The task of

proving E_{g_{0},t} is closable will require proving an integration by parts formula on L(G). The same proof given by Driver [3] for the case of the pinned loop group will hold in our case. The one important key to the proof which requires separate verification in our case is the finite dimensional approximations to the Laplacian and Ricci, which we have already proved in Theorems (3.1.1) and (3.2.1). Although we will not give the complete proof for the integration by parts formula here, we will review how the formula implies the closability of E_{g_{0},t}. It will also be important to explain the combination of probability and geometry that are implicit in the formula.

5.2 The Orthonormal Frame Bundle

We would like to introduce the notion of parallel translation along a Brownian

motion valued on a Lie group. Because of the nature of the Brownian motion, it will be convenient to work in the context of the orthonormal frame bundle.

Let M be a d-dimensional compact, connect Lie group with left invariant met-

ric, ⟨·, ·⟩ (the following discussion will apply also when M is a general manifold). The orthonormal frame bundle, O(M), of M is defined in the following way. For each m ∈ M, let O_{m}(M) be the collection of linear isometries from T_{e}M to T_{m}M (where T_{m}M is the tangent space of M at m). Then let O(M) ≡ ∪_{m∈M}O(M). One may think of O_{m}(M) as being the set of all orthonormal bases of T_{m}M and O(M) as the set of all orthonormal frames on M.

Assume ∇ is a ⟨·, ·⟩-compatible covariant derivative that is torsion skew sym-

metric (i.e., ⟨T ⟨X, Y ⟩, Y ⟩ ≡ 0 for all vector fields X, Y on M, where T is the torsion tensor T ⟨X, Y ⟩ = ∇_{X}Y − ∇_{Y} X − [X, Y] ). Recall that for a function f ∈ C^{∞}(M), the covariant derivative of the differential df is a degree-2 covariant tensor, ∇df, defined by

∇df⟨X, Y ⟩ ≡ X(df⟨Y ⟩) − df⟨∇_{X}Y ⟩.

We define the Laplacian with respect to ∇ by

Δf ≡ tr∇df = ∑_{i=0}^{d}{E_{i}E_{i}f − df⟨∇_{E_{i}}E_{i}⟩ ∀ f ∈ C^{∞}(M),  (5.4)

where {E_{i}}_{i=1}^{d} is a local orthonormal frame on M.

For a smooth path σ : (a, b) → M and a vector field X(s) ∈ T_{σ(s)}M along σ,

we say that X is parallel with respect to ∇ along σ provided

∇_{σ̇(s)}X(s) = 0 ∀ s ∈ (a, b).  (5.5)

Since Brownian motion on M will not be smooth, we need to generalize the meaning of “parallel”.

For each u ∈ O(M), let T_{u}O(M) be the tangent space of O(M) at u. Let

π : T O(M) → O(M) be the projection defined by πX = u for X ∈ T_{u}O(M), and let π̃ : O(M) → M be the projection defined by π̃u ≡ m for u ∈ O_{m}(M). Define the canonical 1-form θ : T O(M) → T_{e}M by

θ⟨X⟩ ≡ (πX)^{−1}π̃_{∗}X ∀ X ∈ O(M).  (5.6)

So if X is of the form X = u′(0) for some smooth curve u in O(M) with u(0) = u ≡ πX, then θ⟨X⟩ = u^{−1}σ′(0) where σ(s) = π̃u(s). In other words, θ⟨X⟩ can be thought of as the “coordinates” of σ′(0) with respect to u (if we think of u as corresponding to an orthonormal basis for T_{m}M where m = π̃u).

Let O(T_{e}M) be the collection of linear isometries from T_{e}M to itself (i.e.,

O(T_{e}M) ≡ O_{e}(M)), and let so(T_{e}M) be the Lie algebra of O(T_{e}M) (i.e., the skewsymmetric linear transformations on T_{e}M). The connection 1-form ω : T O(M) → so(T_{e}M) is defined by

(ω⟨X⟩)⟨a⟩ ≡ u(0)^{−1} \frac{∇}{ds}|_{s=0} (u(s)a) ∀ X ∈ T O(M), a ∈ T_{e}M,  (5.7)

where u is any smooth curve in O(M) with u′(0) = X.

For each a ∈ T_{e}M, we let B_{a} denote the horizontal vector field on T O(M). By

this we mean that for each u ∈ O(M), B_{a}(u) is the unique vector in T_{u}O(M) such that ω⟨B_{a}(u)⟩ = 0 and π̃_{∗}B_{a}(u) = ua. B_{a}(u) is called the horizontal lift of a to u.

5.3 Parallel Translation

We have already defined the meaning of parallel translation in Equation (5.5)

along a smooth path. We are now ready to define the notion of parallel translation along an M-valued Brownian motion. Assume {Σ_{t}}_{t≥0} is an M-valued Brownian motion on a given filtered probability space, (W, {F_{t}}_{t≥0}, F, P), satisfying the usual conditions. By definition, this means that {Σ_{t}}_{t≥0} is an adapted continuous process such that Σ_{0} = e almost everywhere, and

f(Σ_{t}) − f(Σ_{0}) − \frac{1}{2} ∫_{0}^{t}(Δf)(Σ_{τ})dτ  (5.8)

is a martingale ∀ f ∈ C^{∞}(M). We say that an O(M) -valued process, {//_{t}}_{t≥0}, is parallel translation along {Σ_{t}}_{t≥0} provided //_{0} is the identity in O(T_{e}M) , π̃ ∘ //_{t} = Σ_{t} ∀ t ≥ 0, and

∫_{0}^{t} ω⟨δ//_{τ} ⟩ = 0 ∀ t ≥ 0.  (5.9)

One may interpret the integral in (5.9) as follows. Assuming //_{τ} ∈ U for τ ∈ [s_{1}, s_{2}] ⊂ [0, t] and a local chart, (U, x), of O(M), then we can write ω as

ω = ∑_{i=1}^{n} f_{i}dx_{i},

where n ≡ dim(O(M)) and f_{i} : O(M) → so(T_{e}M) for i ∈ {1, …, n}. We then define

∫_{s_{1}}^{s_{2}} ω⟨δ//_{τ} ⟩ ≡ ∑_{i=1}^{n} ∫_{s_{1}}^{s_{2}} f_{i}(//_{τ})δ(x_{i}(//_{τ})).  (5.10)

If //_{τ} lies in more than one chart of O(M), one may use a partition of unity to define the entire integral ∫_{0}^{t} ω⟨δ//_{τ} ⟩ as a combination of integrals of the type in (5.10) (see Chapter 8 in Spivak [20]).

We can see how (5.9) generalizes the notion of parallel translation as given in

Equation (5.5), for if u is a deterministic smooth path in O(M) satisfying (5.9), then we would have

∫_{0}^{t} ω⟨δu(τ)⟩ = ∫_{0}^{t} ω⟨u′(τ)⟩dτ = 0 ∀ t ≥ 0  (5.11)
⇒ ω⟨u′(t)⟩ = 0 ∀ t ≥ 0.

By (5.7) this would imply u(t)^{−1} \frac{∇}{dt}(u(t)a) = 0 ∀ a ∈ T_{e}M. Viewing u as a path in the orthonormal frame bundle, i.e., u(t) corresponds to an orthonormal basis in T_{π̃u(t)}M for each t, we see that (5.11 ) implies that u_{t} is the parallel translation, in the sense of (5.5), of an orthonormal bases u(0) ∈ O(T_{e}M) along the path σ ≡ π̃u in M. Since {Σ_{t}}_{t≥0} is not a smooth process, we see that it is necessary to use a stochastic integral, as in (5.9), to define parallel translation.

5.4 Brownian Motion Via The Development Map

In this section we discuss how Brownian motions on the Lie group, M, and on

the Lie algebra, T_{e}M, are related via the development map and its inverse (sometimes called the “rolling” map). Although we did not refer to it by name, a rolling map was actually used in Theorem 4.0.1 to derive a Brownian motion on the free loop group, L(G).

It will be helpful at this point to give a general outline of what we will do in this

section as it relates to our anticipated integration by parts formula. We will first start with a T_{e}M-valued Brownian motion, {β_{t}}_{t≥0}, and apply the rolling map with respect to a “ left-covariant” derivative to get an M-valued Brownian motion, {Σ_{t}}_{t≥0}. The reason we use the left-covariant and not the Levi-Civita covariant derivative is that it is much easier to solve {Σ_{t}}_{t≥0} in this context since parallel translation turns out to be equivalent to left-translation.

We then use the development map with respect to the Levi-Civita covariant

derivative to derive a different Brownian motion, {b_{t}}_{t≥0}, in T_{e}M. The reason for this will be apparent when we see how the integration by parts formula involves the Levi-Civita Ricci tensor.

Let us now start with a T_{e}M-valued Brownian motion, {β_{t}}_{t≥0}, on the filtered

probability space (W, {F_{t}}_{t≥0}, F, P). Assume {β_{t}}_{t≥0} has covariance

E[⟨β_{t}, h⟩, ⟨β_{τ} , k⟩] = t ∧ τ⟨h, k⟩ ∀ t, τ ≥ 0 and h, k ∈ T_{e}M.

Let ∇^{L} be the unique left-covariant derivative on M, i.e., such that ∇L ; X̃ Ỹ = 0 for all leftinvariant vector fields X̃ and Ỹ on M (see Chapter II, section 3 in Helgason [11]). For a smooth path, σ, in M with σ(0) = e, it is easy to see that for X ∈ T_{e}M, L_{σ(t)∗}X = X̃_{σ(t)} is the parallel translation of X along σ with respect to ∇^{L}. Indeed, if {E_{i}}_{i=1}^{d} is an orthonormal basis of T_{e}M and X = ∑_{i=0}^{d} x^{i}E_{i}, then

∇_{σ̇(t)}^{L}(L_{σ(t)∗}X) = ∇_{σ̇(t)}^{L}(∑_{i=0}^{d} x^{i}(Ẽ_{i})_{σ(t)})
= ∑_{i=0}^{d} x^{i}∇_{σ̇(t)}^{L}(Ẽ_{i})_{σ(t)} = 0.

Let {Σ_{t}}_{t≥0} be the unique solution to the following Stratonovich stochastic

differential equation:

δΣ_{t} = ∑_{i=0}^{d} Ẽ_{i}(Σ_{t})δβ_{t}^{E_{i}}, Σ_{0} = e,  (5.12)

where {E_{i}}_{i=1}^{d} is an orthonormal basis of T_{e}M and β_{t}^{E_{i}} ≡ ⟨E_{i}, β_{t}⟩ for all t ≥ 0. As in Theorem 4.0.1, we interpret (5.12) to mean

f(Σ_{t}) − f(Σ_{0}) = ∑_{i=0}^{d} ∫_{0}^{t} Ẽ_{i}f(Σ_{τ})δβ_{τ}^{E_{i}} ∀ f ∈ C^{∞}(M).

The mapping from {β_{t}}_{t≥0} to {Σ_{t}}_{t≥0} is what we call the rolling map. The existence of a solution to (5.12) is discussed in Chapter V of Ikeda and Watanaba [12]. One may solve such an equation by using Whiteney’s theorem to imbed M into R^{2d+1}. It is the fact that we have used a Stratonovich differential equation that guarantees that the solution to (5.12) will be back in M.

Proposition 5.4.1 {Σ_{t}}_{t≥0} is an M-valued Brownian motion. In other words,

f(Σ_{t}) − f(Σ_{0}) − \frac{1}{2} ∫_{0}^{t}(Δf)(Σ_{τ})dτ

is a martingale for all f ∈ C^{∞}(M), where Δ is defined as in (5.4).

Proof: As we have noted at the end of Chapter 4, since M is unimodular, then

Δ = ∑_{i=1}^{d} Ẽ_{i}^{2}, where {E_{i}}_{i=1}^{d} is again an orthonormal basis for T_{e}M.

By (5.12) and the definition of the Stratonovich integral, for each f ∈ C^{∞}(M)

f(Σ_{t}) − f(Σ_{0}) = ∑_{i=1}^{d}\{∫_{0}^{t}(Ẽ_{i}f)(Σ_{τ})dβ_{τ}^{E_{i}} + \frac{1}{2} ≺ Ẽ_{i}f(Σ), β^{E_{i}} ≻_{t}\},

where ≺ ·, · ≻ denotes quadratic covariation. Note that

∑_{i=1}^{d} ≺ Ẽ_{i}f(Σ), β^{E_{i}} ≻_{t}= ∑_{i=1}^{d} ∫_{0}^{t} d(Ẽ_{i}f(Σ_{τ}))dβ_{τ}^{E_{i}}
= ∑_{i=1}^{d} ∫_{0}^{t} ∑_{j=1}^{d}\{Ẽ_{j}(Ẽ_{i}f)(Σ_{τ})dβ_{τ}^{E_{i}} + \frac{1}{2}d ≺ Ẽ_{j}(Ẽ_{i}f)(Σ), β^{Ej} ≻_{τ} \}dβ_{τ}^{E_{i}}
= ∑_{i=1}^{d} ∑_{j=1}^{d}\{∫_{0}^{t} Ẽ_{j}(Ẽ_{i}f)(Σ_{τ})d ≺ β^{E_{i}}, β^{Ej} ≻_{τ} +\frac{1}{2} ≺≺ Ẽ_{j}(Ẽ_{i}f)(Σ), β^{Ej} ≻, β^{E_{i}} ≻_{t}\},

where we have used (5.12) again for the second equation above. For the third equation, see Theorem 5.7 in Chung and Williams [1]. Since ≺ Ẽ_{j}(Ẽ_{i}f)(Σ), β^{Ej} ≻_{τ} is of bounded variation, then ≺≺ Ẽ_{j}(Ẽ_{i}f)(Σ), β^{Ej} ≻, β^{E_{i}} ≻_{t}= 0. And since {β_{t}}_{t≥0} is a Brownian motion, then ≺ β^{E_{i}}, β^{Ej} ≻_{τ} = δ_{ij}τ (see Theorem 3.16 in Karatzas and Shreve [13]). We thus have

f(Σ_{t}) − f(Σ_{0}) = ∑_{i=1}^{d} ∫_{0}^{t} Ẽ_{i}f(Σ_{τ})dβ_{τ}^{E_{i}} + ∑_{i=1}^{d} \frac{1}{2} ∫_{0}^{t} Ẽ_{i}^{2}f(Σ_{τ})dτ.

Since Ẽ_{i}f(Σ_{τ}) is bounded, then ∫_{0}^{t} Ẽ_{i}f(Σ_{τ})dβ_{τ}^{E_{i}} is a martingale for each. □

Now that we have derived a Brownian motion, {Σ_{t}}_{t≥0}, on M from the Brow-

nian motion {β_{t}}_{t≥0} on T_{e}M, we will derive yet another Brownian motion, {b_{t}}_{t≥0}, on T_{e}M by applying the development map to {Σ_{t}}_{t≥0} with respect to the Levi-Civita covariant derivative.

From now on we will let {//_{t}}_{t≥0} denote parallel translation along {Σ_{t}}_{t≥0}

with respect to the Levi-Civita covariant derivative, which we will denote by ∇. Thus, {//_{t}}_{t≥0} satisfies Equation (5.9). Let {b_{t}}_{t≥0} be the T_{e}M -valued Brownian motion defined by

b_{t} ≡ ∫_{0}^{t} θ⟨δ//_{τ} ⟩ ∀ t ≥ 0,  (5.13)

where θ is the canonical 1-form in (5.6). This mapping sending from {Σ_{t}}_{t≥0} to {b_{t}}_{t≥0} is what we call the development map. With {b_{t}}_{t≥0} defined in this way, one can see that {//_{t}}_{t≥0} satisfies the Stratonovich stochastic differential equation

δ//_{t} = B_{δb_{t}}(//_{t}),  (5.14)

where B_{δb_{t}}(//_{t}) ≡ ∑_{i=1}^{n−1} B_{E_{i}}(//_{t})δb_{t}^{E_{i}} and b_{t}^{E_{i}} ≡ ⟨b_{t}, E_{i}⟩. In a sense, (5.14) says that b_{t} is

the process in T_{e}M whose “ velocity vector” at time t, when parallel translated along Σ, is mapped to the “velocity vector” of Σ at time t. The relationship between {b_{t}}_{t≥0} and {β_{t}}_{t≥0} is given in the theorem below.

Theorem 5.4.2 Define the O(T_{e}M)-valued process U_{t} by

U_{t} ≡ L_{Σ_{t}^{−1}∗}//_{t} .  (5.15)

Then

b_{t} = ∫_{0}^{t} U_{τ}^{−1}δβ_{τ} ,  (5.16)

and

b_{t} = ∫_{0}^{t} U_{τ}^{−1}dβ_{τ} ∀ t ≥ 0.  (5.17)

Also, U_{t} satisfies the Stratonovich differential equation

δU_{t} + D_{δβ_{t}}U_{t} = 0, U_{0} = I_{T_{e}M},  (5.18)

where D is the Levi-Civita derivative, ∇, restricted to T_{e}M.

Proof: We refer to Driver [4] for the proof of (5.18). For (5.16), recall that

b_{t} ≡ ∫_{0}^{t} θ⟨δ//_{τ} ⟩. By (5.6) and the fact that Σ_{t} = π̃ ∘ //_{t} , we get

b_{t} = ∫_{0}^{t}(πδ//_{τ})^{−1}π̃_{∗}δ//_{τ}
= ∫_{0}^{t} //_{τ}^{−1}δΣ_{τ} .

Applying (5.12) gives

b_{t} = ∫_{0}^{t} //_{τ}^{−1}{∑_{i=0}^{d} Ẽ_{i}(Σ_{τ})δβ_{τ}^{E_{i}}}
= ∫_{0}^{t} //_{τ}^{−1}L_{Σ_{τ} ∗}{∑_{i=0}^{d} E_{i}(Σ_{τ})δβ_{τ}^{E_{i}}}
= ∫_{0}^{t} U_{τ}^{−1}δβ_{τ} ,

which proves (5.16).

For (5.17), the definition of the Stratonovich stochastic integral in (5.16) gives

b_{t} = ∫_{0}^{t} U_{τ}^{−1}dβ_{τ} + \frac{1}{2} ≺ U^{−1}, β ≻_{t}
= ∫_{0}^{t} U_{τ}^{−1}dβ_{τ} + \frac{1}{2} ∫_{0}^{t} d(U_{τ}^{−1})dβ_{τ} .

We claim that {∫_{0}^{t} U_{τ}^{−1}dβ_{τ} }_{t≥0} is a Brownian motion. Since {b_{t}}_{t≥0} is also a

Brownian motion in T_{e}M and ≺ U^{−1}, β ≻_{t} is a process of bounded variation, it must be that \frac{1}{2} ≺ U^{−1}, β ≻_{t}= 0 for all t ≥ 0, and thus b_{t} = ∫_{0}^{t} U_{τ}^{−1}dβ_{τ}.

To see that {∫_{0}^{t} U_{τ}^{−1}dβ_{τ} }_{t≥0} is a O(T_{e}M)-valued Brownian motion, let (A_{τ}^{ij}) be

the matrix corresponding to U_{τ}^{−1} with respect to a fixed basis of O(T_{e}M) and let β_{τ}^{i} be the ith component of β_{τ} with respect to this basis. Since L_{Σ_{τ}^{−1}∗} and //_{τ} are isometries, then by definition of U_{τ} it is clear that (A_{τ}^{ij}) is orthogonal. We may thus calculate the quadratic covariation of the ith and kth components of ∫_{0}^{t} U_{τ}^{−1}dβ_{τ} as follows:

≺ ∫_{0}^{·} A_{τ}^{ij}dβ_{τ}^{j}, ∫_{0}^{·} A_{τ}^{kl}dβ_{τ}^{l} ≻_{t}= ∫_{0}^{t} A_{τ}^{ij}A_{τ}^{kl}d ≺ β^{j}, β^{l} ≻_{τ}
= ∫_{0}^{t} A_{τ}^{ij}A_{τ}^{kl}δ_{jl}dτ
= ∫_{0}^{t} A_{τ}^{ij}(A_{τ}^{−1})^{jk}dτ
= δ_{ik}t.

By Lévy’s martingale characterization of Brownian motion (see Theorem 3.16 in [13]), it follows that {∫_{0}^{t} U_{τ}^{−1}dβ_{τ} }_{t≥0} is indeed a Brownian motion. □

Note that since U_{τ}^{−1}U_{τ} = I_{T_{e}M}, Stratonovich integration by parts gives

δ(U_{τ}^{−1}U_{τ}) = (δU_{τ}^{−1})U_{τ} + U_{τ}^{−1}δU_{τ} = 0.

Thus,

δU_{τ}^{−1} = −U_{τ}^{−1}(δU_{τ})U_{τ}^{−1}
= U_{τ}^{−1}(D_{δβ_{τ}} U_{τ})U_{τ}^{−1}
= U_{τ}^{−1}D_{δβ_{τ}} ,

where we have used (5.18) for the last equation. This gives

∫_{0}^{t} d(U_{τ}^{−1})dβ_{τ} = ∫_{0}^{t} δ(U_{τ}^{−1})dβ_{τ} − \frac{1}{2} ∫_{0}^{t} d ≺ U^{−1}, 1 ≻_{τ} dβ_{τ}
= ∫_{0}^{t} δ(U_{τ}^{−1})dβ_{τ}
= ∫_{0}^{t} U_{τ}^{−1}D_{δβ_{τ}} dβ_{τ}
= ∫_{0}^{t} U_{τ}^{−1}D_{dβ_{τ}} dβ_{τ} − \frac{1}{2} ∫_{0}^{t} U_{τ}^{−1}D_{d≺β,1≻_{τ}} dβ_{τ}
= ∫_{0}^{t} U_{τ}^{−1}D_{dβ_{τ}} dβ_{τ}

= ∫_{0}^{t} U_{τ}^{−1} ∑_{i,j=1}^{d}(D_{E_{i}}E_{j})dβ_{τ}^{E_{i}}dβ_{τ}^{E_{j}}
= ∫_{0}^{t} U_{τ}^{−1} ∑_{i=1}^{d}(D_{E_{i}}E_{i})dτ.

By the last part of the proof of Theorem 5.4.2, we know that ≺ U^{−1}, β ≻_{t}= 0 for all t ≥ 0, so by the above calculation,

∫_{0}^{t} U_{τ}^{−1} ∑_{i=1}^{d}(D_{E_{i}}E_{i})dτ = 0 ∀ t ≥ 0.

Since U_{τ}^{−1} is obviously invertible, then this implies that ∑_{i=1}^{d}(D_{E_{i}}E_{i}) = 0. We may thus write the Laplacian in (5.4) as

Δf = ∑_{i=1}^{d} Ẽ_{i}(Ẽ_{i}f),

which agrees with the Laplace Beltrami operator as in Proposition 5.4.1.

5.5 Integration By Parts

We are now ready to give our integration by parts formula for left invariant

vector fields on L(G). Let β and Σ be the Brownian motions on L(𝔤) and L(G) as in Theorem 4.0.1. For what follows, if h ∈ H(𝔤), we define (h, β_{t}) to mean a continuous version of

lim_{n→∞} α_{h_{n}}(β_{t}),  (5.19)

where α is defined as in (4.1), {h_{n}} ⊂ H^{BV} with h_{n} → h in H(𝔤), and the limit in (5.19) is in L^{2} with respect to the probability measure P on the space W on which β is defined.

Theorem 5.5.1 Let t > 0 and h ∈ H(𝔤). Assume l : [0, t] → R is an absolutely continuous function with l(0) = 0, l(t) = 1, and ∫_{0}^{t} l̇^{2}(τ)dτ < ∞. Let Ric be the Levi-Civita Ricci tensor on H(𝔤) as defined in (2.21). Then for all f ∈ FC^{∞}(L(G)),

E[(h̃f)(Σ_{t})] = E[f(Σ_{t}) ∫_{0}^{t}(\{l̇(τ) − \frac{1}{2}l(τ)Ric\}H(τ), \overleftarrow{dβ}_{τ})],  (5.20)

where H(t) H(𝔤) is the unique H(𝔤)-valued process that solves the backwards Stochastic differential equation

h − H(s) + ∫_{s}^{t} ∇\overleftarrow{dβ}_{τ} H(τ) + \frac{1}{2} ∫_{0}^{t} ΔH(τ)dτ = 0.  (5.21)

Here, ∇ is our Levi-Civita covariant derivative on H(𝔤) (as in Definition 2.1.13 and Δ is the Laplacian Δ ≡ ∑_{k∈S} ∇_{k}^{2} (as in Definition 3.0.9. By abuse of notation, we take the inner product in the right side of (5.20) to mean

(\{l̇(τ) − \frac{1}{2}l(τ)Ric\}H(τ), \overleftarrow{dβ}_{τ}) ≡ (l̇(τ)H(τ), \overleftarrow{dβ}_{τ}) − \frac{1}{2}l(τ)Ric⟨H(τ), \overleftarrow{dβ}_{τ} ⟩.

Sketch of Proof: We will discuss the key steps in the proof as it is presented

in Driver [3]. Assume P = {0 = s_{0} < s_{1} < ⋯ < s_{n−1} < 1} is a partition of [0, 1], and f = F ∘ π_{P}, where F ∈ C^{∞}(G^{P}). Let (Σ_{P})_{t} ≡ π_{P} ∘ Σ_{t} = (Σ_{t}(s_{0}), …, Σ_{t}(s_{n−1})) and (β_{P})_{t} ≡ π̃_{P} ∘ β_{t} = (β_{t}(s_{0}), …, β_{t}(s_{n−1})). As in Theorem 4.0.1, Σ_{P} and β_{P} are related by the equation

(Σ_{P})_{δt} = ∑_{j=0}^{dn} Ẽ_{j}((Σ_{P})_{t})δ(β_{P})_{t}^{E_{j}}
= ∑_{i=0}^{n} ∑_{j=0}^{dn} E_{j}^{(i)}((Σ_{P})_{t})⟨β_{δt}(s_{i}), E_{j}⟩,

where d = dim(G) and {E_{i}}_{i=1}^{d} is an orthonormal basis for T_{e}M.

For the compact, finite dimensional Lie group, M ≡ G^{P}, the following finite

dimensional version of (5.21) holds by Corollary (6.5) in Driver [4]:

E[(h̃f)(Σ_{t})] = E[(h̃_{P}F)((Σ _{P})_{t})] = E[f((Σ_{P})_{t}) ∫_{0}^{t}(\{l̇(τ) − \frac{1}{2}l(τ)Ric_{P}\}H_{P}(τ), \overleftarrow{d(β_{P})}_{τ})], (5.22)

where h_{P} ≡ π_{P}h, and Ric_{P} is the Ricci tensor with respect to the Levi-Civita covariant derivative, ∇^{P}, on 𝔤^{P}. We define the 𝔤^{P} -valued process by

(H_{P})_{τ} ≡ U_{P}(τ, t)h_{P} ≡ U_{P}(τ)U_{P}(t)^{−1}h_{P},

where U_{P}(t) is the O(𝔤^{P}) − valued process,

U_{P}(t) ≡ L_{(Σ_{P})_{t}^{−1}∗}(//_{P})_{t},

satisfying the Stratonovich differential equation

dU_{P}(t) + ∇_{δ(β_{P})_{t}}^{P}U_{P}(t) = 0, U_{P}(0) = I_{𝔤^{P}},

and (//_{P})_{t} is parallel translation along (Σ_{P})_{t}.

We should note that the proof for (5.22) relies on the use of the 𝔤^{P}-valued

Brownian motion {(b_{P})_{t}}_{t≥0} resulting from the development map applied to {(Σ_{P})_{t}}_{t≥0} as in (5.13).

The next key to the proof of (5.20) is passing to the limit as |P| → 0 in (5.22).

This requires a detailed argument for which we refer to the one offered by Driver [3]. The reason the same argument applies in our case is due to the crucial fact that our finite dimensional Ricci tensors and Laplacians converge to their infinite dimensional counterparts as we have shown in Theorems (3.1.1) and (3.2.1). □

5.6 Closability of the Quadratic Form

Recall that the quadratic form we are interested in for the logarithmic Sobolev

inequality is given by

E_{g_{0},t}(f, g) ≡ ∫_{L(G)}(∇f(x), ∇g(x))_{H(𝔤)}ν_{t}(g_{0}, dx)

for each t ≥ 0 and g_{0} ∈ L(G), where ∇ : D(∇) = FC^{∞}(L(G)) → L^{2}(L(G), ν_{t}(g_{0}, ·); H(𝔤)) is the gradient operator satisfying the condition in (5.2). As in (4.6), ν_{t}(e, ·) is the law at time t of the Brownian motion {Σ_{t}}_{t≥0} constructed in (5.12) and ν_{t}(g_{0}, ·) = L_{g_{0}∗}ν_{t}(e, ·).

Theorem 5.6.1 For each t > 0 and g_{0} ∈ L(G), E_{g_{0},t} is closable, where D(E_{g_{0},t}) = FC^{∞}(L(G)).

Proof: It is sufficient to prove E_{t} ≡ E_{e,t} is closable for every t ≥ 0 since for each

g_{0} ∈ L(G),

E_{g_{0},t}(f, g) = ∫_{L(G)}(∇f(x), ∇g(x))_{H(𝔤)}ν_{t}(g_{0}, dx)
= ∫_{L(G)}(∇f(L_{g_{0}}x), ∇g(L_{g_{0}}x))_{H(𝔤)}ν_{t}(e, dx)
= ∫_{L(G)}(∇(f ∘ L_{g_{0}})(x), ∇(g ∘ L_{g_{0}})(x))_{H(𝔤)}ν_{t}(e, dx)
= E_{t}(f ∘ L_{g_{0}}, g ∘ L_{g_{0}})

for all f, g ∈ D(E_{t}) = FC^{∞}(L(G)). Thus, if we knew E_{t} were closable, then given {f_{n}} ⊆ D(E_{t}) with ∥f_{n}∥_{L^{2}(L(G),ν_{t}(g_{0},·))} → 0 and E_{g_{0},t}(f_{n} − f_{m}) → 0 as m, n → ∞, then

∥f_{n} ∘ L_{g_{0}}∥_{L^{2}(L(G),ν_{t}(e,·))} → 0
and E_{g_{0},t}(f_{n} ∘ L_{g_{0}} − f_{m} ∘ L_{g_{0}}) → 0 as m, n → ∞.

So E_{g_{0},t}(f_{n}) = E_{t}(f_{n} ∘ L_{g_{0}}) → 0 as n → ∞.

We next show that ∇ has a densely defined adjoint. Using our integration by

parts formula in (5.20) with l(τ) ≡ \frac{τ}{t}, for example, we see that for all h ∈ H(𝔤) and f, g ∈ FC^{∞}(L(G)),

⟨∇(fg), h⟩_{L^{2}(L(G),ν_{t}(e,·);H(𝔤))} = ∫_{L(G)}(∇(fg)(x), h)_{H(𝔤)}ν_{t}(e, dx)
= ∫_{L(G)} h̃(fg)(x)ν_{t}(e, dx)
= E[h̃(fg)(Σ_{t})]
= E[(fg)(Σ_{t})\frac{1}{t} ∫_{0}^{t}(\{I − \frac{1}{2}τRic\}H(τ), \overleftarrow{dβ}_{τ})]
= E[(fg)(Σ_{t})\frac{1}{t}E[∫_{0}^{t}(\{I − \frac{1}{2}τRic\}H(τ), \overleftarrow{dβ}_{τ}) | σ(Σ_{t})]]
= E[(fg)(Σ_{t})φ_{h}(Σ_{t})],

where φ_{h} : L(G) → R is a Borel measurable function such that

φ_{h}(Σ_{t}) = \frac{1}{t}E[∫_{0}^{t}(\{I − \frac{1}{2}τRic\}H(τ), \overleftarrow{dβ}_{τ})_{H(𝔤)} | σ(Σ_{t})] a.s..

Thus, letting L^{2} ≡ L^{2}(L(G), ν_{t}(e, ·); H(𝔤)), we have

⟨∇f, gh⟩_{L^{2}} + ⟨∇g, fh⟩_{L^{2}} = ⟨∇(fg), h⟩_{L^{2}}
= ∫_{L(G)}(fg)(x)φ_{h}(x)ν_{t}(e, dx)
= ⟨f, gφ_{h}⟩_{L^{2}(L(G),ν_{t}(e,·))},

and thus

⟨∇f, gh⟩_{L^{2}} = ⟨f, gφ_{h}⟩_{L^{2}(L(G),ν_{t}(e,·))} − ∫_{L(G)} f(x)h̃g(x)ν_{t}(e, dx)
= ⟨f, gφ_{h} − h̃g⟩_{L^{2}(L(G),ν_{t}(e,·))}.  (5.23)

Letting FC^{∞}(L(G), H(𝔤)) ≡ {gh : g ∈ L(G), h ∈ H(𝔤)}, we see by (5.23) that for gh ∈ FC^{∞}(L(G), H(𝔤))

∇^{∗}(gh) = gφ_{h} − h̃g.

Thus, FC^{∞}(L(G), H(𝔤)) ⊂ D(∇^{∗}).

We next show that FC^{∞}(L(G), H(𝔤)) is dense in L^{2} = L^{2}(L(G), ν_{t}(e, ·); H(𝔤))

by showing that FC^{∞}(L(G), H(𝔤))^{⊥} = {0} (see Lemma 3.3-7 in Kreyszig [14]). Indeed, suppose that J ∈ FC^{∞}(L(G), H(𝔤)) and (gh, J)_{L^{2}} = 0 for g ∈ L(G) and h ∈ H(𝔤). Then

∫_{L(G)} g(x)(h, J(x))_{H(𝔤)}ν_{t}(e, dx) = 0.

Since g was arbitrary, implies that for all h ∈ H(𝔤), (h, J(x))_{H(𝔤)} for almost all x. Thus, J(x) = 0 for almost all x since we can write

J(x) = ∑_{h∈S}(h, J(x))_{H(𝔤)}

where S is an orthonormal (countable) basis of S.

Finally, by Lemma 5.1.5, we can conclude that ∇ is closable. Lemma 5.1.6 then

implies that E_{t} is closable. □

The Logarithmic Sobolev

Inequality

We are now ready to state the finite dimensional version of our logarithmic

Sobolev inequality. For any F ∈ C^{∞}(G^{P}), let

P_{t}F (g_{0P}) ≡ ∫_{G^{P}} F (g_{P})ν_{t}^{P}(g_{0P}, dg_{P}).

Theorem 6.0.1 Let C be the constant in Theorem 2.3.11 such that Ric⟨h, h⟩ ≥ −C(h, h) ∀ h ∈ H(𝔤). Then ∃ M > 0 such that for a fixed partition, P, of [0, 1] and for t ≥ 0,

P_{t}(F^{2} log F^{2}) ≤ \frac{2}{C_{0}} (e^{C_{0}t} − 1)P_{t}(∥∇_{P}F ∥_{𝔤^{P}}^{2}) + P_{t}(F^{2}) log(P_{t}(F^{2}))  (6.1)

on G^{P}, where C_{0} ≡ C + M and ˜∇_{P}F is the gradient of F on G^{P} with respect to (·, ·)_{P}. By convention, 0 log 0 ≡ 0.

Proof: Recall that by Theorem 3.2.1, for each partition P, there exists an

ϵ(P) > 0 such that

|Ric⟨h, h⟩ − Ric_{P}⟨h, h⟩| ≤ ϵ(P) ∥h∥_{H(𝔤)}^{2} ∀ h ∈ H_{P}(𝔤).  (6.2)

It is not difficult to see from the proof of Theorem 3.2.1 that ϵ(P) is uniformly bounded, i.e., ∃ M > 0 such that ϵ(P) ≤ M for all partitions P of [0, 1]. From (6.2), we thus have for each partition P,

Ric_{P}⟨h, h⟩ ≥ Ric⟨h, h⟩ − ϵ(P) ∥h∥_{H(𝔤)}^{2}
≥ −C ∥h∥_{H(𝔤)}^{2} − M ∥h∥_{H(𝔤)}^{2} .

Hence,

Ric_{P}⟨h, h⟩ ≥ −C_{0} ∥h∥_{H(𝔤)}^{2} ,

where C_{0} ≡ C + M > 0.

Let ∇^{P} be the Levi-Civita covariant derivative on H_{P}(𝔤) as described in Def-

inition 3.0.8. Let ˜∇^{P} denote the Levi-Civita covariant derivative on 𝔤^{P} with respect to the metric (·, ·)_{P} defined in (3.1). Since π̃_{P} : H_{P}(𝔤) → 𝔤^{P} is a Lie algebra isometric isomorphism, by Propositions 3.0.5 and 3.0.7 , it easy to check that for all A⃗ and B⃗ in 𝔤^{P},

˜∇P ; A⃗B⃗ = π̃_{P}∇_{(π̃_{P}^{−1}A⃗)}^{P}(π̃_{P}^{−1}B⃗).

Now let R_{𝔤^{P}} be the Levi-Civita curvature tensor and Ric_{𝔤^{P}} be the Levi-Civita Ricci tensor with respect to (·, ·)_{P} on G^{P}. We can see then that for A⃗ ∈ 𝔤^{P},

Ric_{𝔤^{P}}⟨A⃗, A⃗⟩
= ∑_{B⃗∈𝔤_{0}^{P}}(R_{𝔤^{P}}⟨A⃗, B⃗⟩B⃗, A⃗)_{P}
= ∑_{B⃗∈𝔤_{0}^{P}}(π̃_{P}∇_{(π̃_{P}^{−1}A⃗)}^{P}(π̃_{P}^{−1}π̃_{P}∇_{(π̃_{P}^{−1}B⃗)}^{P}(π̃_{P}^{−1}B⃗)) − π̃_{P}∇_{(π̃_{P}^{−1}B⃗)}^{P}(π̃_{P}^{−1}π̃_{P}∇_{(π̃_{P}^{−1}A⃗)}^{P}(π̃_{P}^{−1}B⃗))
−π̃_{P}∇_{(π̃_{P}^{−1}[A⃗,B⃗])}^{P}(π̃_{P}^{−1}B⃗), A⃗)_{P}
= ∑_{B⃗∈𝔤_{0}^{P}}(∇_{(π̃_{P}^{−1}A⃗)}^{P}(∇_{(π̃_{P}^{−1}B⃗)}^{P}(π̃_{P}^{−1}B⃗)) − ∇_{(π̃_{P}^{−1}B⃗)}^{P}(∇_{(π̃_{P}^{−1}A⃗)}^{P}(π̃_{P}^{−1}B⃗))
−∇_{(π̃_{P}^{−1}[A⃗,B⃗])}^{P}(π̃_{P}^{−1}B⃗), π̃_{P}^{−1}A⃗)_{H(𝔤)}
= ∑_{k∈S_{P}}(∇_{(π̃_{P}^{−1}A⃗)}^{P}(∇_{h}^{P}h) − ∇_{h}^{P}(∇_{(π̃_{P}^{−1}A⃗)}^{P}h) − ∇_{[π̃_{P}^{−1}A⃗,h]_{P}}^{P} h, π̃_{P}^{−1}A⃗)_{H(𝔤)}
= Ric_{P}⟨π̃_{P}^{−1}A⃗, π̃_{P}^{−1}A⃗⟩.

Thus

Ric_{𝔤^{P}}⟨A⃗, A⃗⟩ = Ric_{P}⟨π̃_{P}^{−1}A⃗, π̃_{P}^{−1}A⃗⟩
≥ −C_{0}(π̃_{P}^{−1}A⃗, π̃_{P}^{−1}A⃗)_{H(𝔤)}
= −C_{0}(A⃗, A⃗)_{P}.

By left-translation invariance of Ric_{𝔤^{P}} and (·, ·)_{P}, it follows that for all partitions P,

Ric_{𝔤^{P}}⟨A⃗, A⃗⟩ ≥ −C_{0}(A⃗, A⃗)_{P} ∀ A⃗ ∈ T G^{P}.

Since G^{P} is a unimodular Lie group, then the inequality in (6.1) follows by

Theorem 2.9 in Driver and Lohrenz [5] and is originally due to Bakry and Ledoux. □

Since the constant, C_{0}, in Theorem 6.0.1 is independent of the partition P,

then the logarithmic Sobolev inequality holds in the context of L(G) as stated in the theorem below.

Theorem 6.0.2 (Logarithmic Sobolev Inequality on Cylinder Functions) Let C_{0} be the constant above and let g_{0} ∈ L(G) be fixed. For all f ∈ FC^{∞}(L(G)) and t > 0,

∫_{L(G)} f^{2} log f^{2}dμ ≤ \frac{2}{C_{0}} (e^{C_{0}t} − 1) ∫_{L(G)} ∥∇f∥_{H(𝔤)}^{2} dμ + ∫_{L(G)} f^{2}dμ log(∫_{L(G)} f^{2}dμ), (6.3)

where μ ≡ ν_{t}(g_{0}, ·).

Proof: The inequality in (6.3) follows from Theorem 6.0.1 with the following

observations. First, note that for f ∈ FC^{∞}(L(G)) of the form f = F ∘π_{P} , F ∈ C^{∞}(G^{P}),

∫_{L(G)} f^{2} log f^{2}dμ = ∫_{L(G)} F^{2}(π_{P}g) log F^{2}(π_{P}g)ν_{t}(g_{0}, dg)
= ∫_{W} F^{2}(π_{P} ∘ L_{g_{0}} ∘ Σ_{t}) log F^{2}(π_{P} ∘ L_{g_{0}} ∘ Σ_{t})dP
= ∫_{W} F^{2}(L_{g_{0P}} ∘ (Σ_{P})_{t}) log F^{2}(L_{g_{0P}} ∘ (Σ_{P})_{t})dP
= ∫_{G^{P}} F^{2}(g_{P}) log F^{2}(g_{P})ν_{t}^{P}(g_{0P}, dg_{P}).  (6.4)

Similarly,

∫_{L(G)} f^{2}dμ log(∫_{L(G)} f^{2}dμ) = ∫_{G^{P}} F^{2}(g_{P})ν_{t}^{P}(g_{0P}, dg_{P}) log(∫_{G^{P}} F^{2}(g_{P})ν_{t}^{P}(g_{0P}, dg_{P})). (6.5)

Next, note that using notation from the proof of Lemma 4.0.4, we have

∫_{L(G)} ∥∇f∥_{H(𝔤)}^{2} dμ = ∫_{L(G)} ∥ ∑_{A∈𝔤_{0}} ∑_{i=0}^{n−1}(A^{(i)}F (π_{P}g))G(s_{i}, ·)A∥_{H(𝔤)}^{2} ν_{t}(g_{0}, dg)
= ∫_{G^{P}} ∥ ∑_{A∈𝔤_{0}} ∑_{i=0}^{n−1}(A^{(i)}F (g_{P}))G(s_{i}, ·)A∥_{H(𝔤)}^{2} ν_{t}^{P}(g_{0P}, dg_{P})
= ∫_{G^{P}} ∥ ∑_{A∈𝔤_{0}} ∑_{i=0}^{n−1}(A^{(i)}F (g_{P}))G_{P}(s_{i})A∥_{𝔤^{P}}^{2} ν_{t}^{P}(g_{0P}, dg_{P}),

where, for the last equation, we have used the fact that π_{P} : H_{P}(𝔤) → 𝔤^{P} is an isometry. But for any h = (h_{0}, h_{1}, …, h_{n−1}) ∈ 𝔤^{P},

( ∑_{A∈𝔤_{0}} ∑_{i=0}^{n−1}(A^{(i)}F (g_{P}))G_{P}(s_{i})A, h)_{𝔤^{P}} = ∑_{A∈𝔤_{0}} ∑_{i=0}^{n−1}(A^{(i)}F (g_{P}))⟨A, h_{i}⟩
= h̃F (g_{P}).

Hence, ∑_{A∈𝔤_{0}} ∑_{i=0}^{n−1}(A^{(i)}F (g _{P}))G_{P}(s_{i})A = ∇_{P}F (g_{P}). We may thus write

∫_{L(G)} ∥∇f∥_{H(𝔤)}^{2} dμ = ∫_{G^{P}} ∥∇_{P}F (g_{P})∥_{𝔤^{P}}^{2} ν_{t}^{P}(g_{0P}, dg_{P}).  (6.6)

Using the equivalences in Equations (6.4), (6.5), and (6.6). as substitutions in

(6.1) gives us (6.3). □

Since we have shown that E_{g_{0},t} is closable for all g_{0} ∈ L(G) and t > 0, we can

extend the inequality in Theorem 6.0.2 to the Hilbert space D(\overline{E}_{g_{0},t}) where \overline{E}_{g_{0},t} is the closed extension of E_{g_{0},t} as in (5.1).

Theorem 6.0.3 (General Logarithmic Sobolev Inequality)Let C_{0} be the constant in Theorem 6.0.1 and let g_{0} ∈ L(G) be fixed. For all t > 0 and f ∈ D(\overline{E}_{g_{0},t}),

∫_{L(G)} f^{2} log f^{2}dμ ≤ \frac{2}{C_{0}} (e^{C_{0}t} − 1)\overline{E}_{g_{0},t}(f) + ∫_{L(G)} f^{2}dμ log(∫_{L(G)} f^{2}dμ), (6.7)

where μ ≡ ν_{t}(g_{0}, ·).

Proof: Let t > 0 and f ∈ D(\overline{E}_{g_{0},t}). By our discussion in Section 5.1, there

exists a (·, ·)_{1}-Cauchy sequence, {f_{n}}, in D(E_{g_{0},t}) = FC^{∞}(L(G)) such that f_{n} → f in L^{2}(L(G), ν_{t}(g_{0}, ·)) and \overline{E}_{g_{0},t}(f_{n}) = lim_{n→∞} E_{g_{0},t}(f_{n}). Since f_{n} is L^{2}-convergent, there exists a subsequence, {f_{n_{k}}}, which converges pointwise to f a.e.. Let −M, with M > 0, be a lower bound for the function x ↦ x log x on (0, ∞) so that M + f_{n_{k}}^{2} log f_{n_{k}}^{2} ≥ 0 for all k. Fatou’s Lemma then implies

∫_{L(G)}{M + f^{2} log f^{2}}dμ ≤ lim inf_{k→∞} ∫_{L(G)}{M + f_{n_{k}}^{2} log f_{n_{k}}^{2}}dμ

and thus

∫_{L(G)} f^{2} log f^{2}dμ ≤ lim inf_{k→∞} ∫_{L(G)} f_{n_{k}}^{2} log f_{n_{k}}^{2}dμ.

Using Theorem 6.0.2, this give us

∫_{L(G)} f^{2} log f^{2}dμ ≤ lim inf_{k→∞} ∫_{L(G)} f_{n_{k}}^{2} log f_{n_{k}}^{2}dμ
≤ lim_{k→∞}\{\frac{2}{C_{0}} (e^{C_{0}t} − 1)E_{g_{0},t}(f_{n_{k}})
+ ∫_{L(G)} f_{n_{k}}^{2}dμ log(∫_{L(G)} f_{n_{k}}^{2}dμ)}
= \frac{2}{C_{0}} (e^{C_{0}t} − 1)\overline{E}_{g_{0},t}(f) + ∫_{L(G)} f^{2}dμ log(∫_{L(G)} f^{2}dμ). □

In conclusion, one may refer to Davies [2] or Gross [10] for some general proper-

ties of logarithmic Sobolev inequalities. In particular, Gross discusses the relationships between logarithmic Sobolev inequalities, such as the one in (6.7) above, and the unique self-adjoint operator, H_{g_{0},t}, associated to the closed form, \overline{E}_{g_{0},t}, appearing in the inequality (i.e., \overline{E}_{g_{0},t}(f, g) = (H_{g_{0},t}f, g)_{L^{2}} for all f, g ∈ D(\overline{E}_{g_{0},t})). For example, by the Rothaus-Simon mass gap theorem, (6.7) implies that

∥f∥_{L^{2}} − (f, 1)_{L^{2}}^{2} ≤ \frac{e^{C_{0}t} − 1}{C_{0}} \overline{E}_{g_{0},t}(f)

for all f ∈ D(\overline{E}_{g_{0},t}). This implies that the spectrum of H_{g_{0},t} is contained in \{0\} ∩ [\frac{C_{0}}{e^{C_{0}t}−1}, ∞). Gross also discusses an equivalence between the logarithmic Sobolev inequality and a particular hypercontractivity property of the operator semigroup, P (T) ≡ e^{−T H_{g_{0},t}}, associated to H_{g_{0},t}.

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