The strongly local part of nonlinear Dirichlet structures and its energy measures

Abstract

This paper substantially extends the results of Sasaya (2026, JFA): we propose a framework for nonlinear generalizations of regular Dirichlet forms and prove that each energy in this framework has a largest strongly local part. We also prove the existence and uniqueness of canonical energy measures of this part, which satisfy convexity and locality properties and, under additional assumptions, certain chain rules in one variable. In particular, our framework requires neither $p$-homogeneity nor strong locality. Moreover, under a Ball--Carlen--Lieb-type upper bound, we prove more detailed chain rules for the directional derivatives of these energy measures. Our framework is applicable both to $p$-energy forms on fractals and to a class of symmetric nonlinear Dirichlet forms in the sense of Cipriani and Grillo~\cite{CG2003}, including those of Orlicz--Sobolev and variable-exponent Sobolev type.

arXiv comment72pages, 2 figures. Comments are welcome. The author would be grateful for help identifying the ICMAT workshop participant mentioned in the acknowledgments