The Normal Contraction Property for Non-Bilinear Dirichlet Forms

Abstract

We analyse the class of convex functionals \(\mathcal {E}\) over L2(X,m) for a measure space (X,m) introduced by Cipriani and Grillo (J. Reine Angew. Math. 562, 201–235 2003) and generalising the classic bilinear Dirichlet forms. We investigate whether such non-bilinear forms verify the normal contraction property, i.e., if \(\mathcal {E}(\phi \circ f) \leq \mathcal {E}(f)\) for all f ∈L2(X,m), and all 1-Lipschitz functions \(\phi : \mathbb {R} \to \mathbb {R}\) with ϕ(0) = 0. We prove that normal contraction holds if and only if \(\mathcal {E}\) is symmetric in the sense \(\mathcal {E}(-f) = \mathcal {E}(f),\) for all f ∈L2(X,m). An auxiliary result, which may be of independent interest, states that it suffices to establish the normal contraction property only for a simple two-parameter family of functions ϕ.