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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2022
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2205.02928 |
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Table of Contents:
- 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 $\mathcal E(ϕ\circ f) \leq \mathcal E(f)$ for all $f \in \mathrm{L}^2(X,m)$, and all 1-Lipschitz functions $ϕ: \mathbb R \to \mathbb R$ 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 $f \in \mathrm{L}^2(X,m).$ 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 $ϕ$.