Tohoku Mathematical Journal
2004
September
SECOND SERIES VOL. 56, NO. 3

Tohoku Math. J.
56 (2004), 327-340

Title SUPERPOSITION OPERATORS ON DIRICHLET SPACES

Dedicated to Professor Ronald Getoor on his seventy-fifth birthday
Author Patrick J. Fitzsimmons

(Received November 11, 2002, revised October 3, 2003)
Abstract. In the context of a strongly local Dirichlet space we show that if a function mapping the real line to itself (and fixing the origin) operates by composition on the left to map the Dirichlet space into itself, then the function is necessarily locally Lipschitz continuous. If, in addition, the Dirichlet space contains unbounded elements, then the function must be globally Lipschitz continuous. The proofs rely on a co-area formula for condenser potentials.

2000 Mathematics Subject Classification. Primary 31C25; Secondary 60J45, 46E35.

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