The normal contraction property for non-bilinear Dirichlet forms

Abstract

We analyse the class of convex functionals $\mathcal{E}$ over ${\mathrm{L}}^2(X,m)$ for a measure space $(X,m)$ introduced by Cipriani and Grillo and generalising the classic bilinear Dirichlet forms. We investigate whether such non-bilinear forms verify the normal contraction property, i.e., if $\mathcal{E}(\varphi \circ f)\le \mathcal{E}(f)$ for all $f\in {\mathrm{L}}^2(X,m)$, and all 1-Lipschitz functions $\varphi :\mathbb{R}\to \mathbb{R}$ with $\varphi (0)=0$. 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 $f\in {\mathrm{L}}^2(X,m).$ 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 $\varphi$.

arXiv comment21 pages, 8 figures, improved presentation

References in corpus (4)

• Metric measure spaces with Riemannian Ricci curvature bounded from below
• Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
• Bakry-Émery curvature-dimension condition and Riemannian Ricci curvature bounds
• Density of Lipschitz functions and equivalence of weak gradients in metric measure spaces