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.

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November 4, 2002; revised July 25, 2003