HRMs and Strongly Supermedian Kernels
### Superposition Operators on Dirichlet Spaces

#### P.J. Fitzsimmons

In the context of a strongly local Dirichlet form (*E,D*),
we show that if *K* : **R** --> **R** is a measurable function with *K* (0)=0
such that *K(u)* (functional composition) is an element of *D*
whenever *u* is an element of *D*, then *K* is necessarily locally
Lipschitz continuous. If, in addition, *D* contains unbounded elements,
then *K* must be globally Lipschitz continuous. The proofs rely on a
co-area formula for condenser potentials.

A hard copy of this manuscript is available from the
author upon request.

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November 4, 2002; revised July 25, 2003