Korevaar-Schoen and heat kernel characterizations of Sobolev and BV spaces on local trees

TL;DR

This paper develops a theory of Sobolev and BV spaces on local trees using intrinsic geodesic structures, characterizations via energy functionals and heat kernels, and explores applications like interpolation, critical exponents, and heat semigroup bounds.

Contribution

It introduces a new framework for Sobolev and BV spaces on local trees, with dual characterizations and several applications, extending analysis on these metric spaces.

Findings

Characterization of Sobolev and BV spaces via Korevaar-Schoen energies.

Heat kernel-based characterization of function spaces.

Establishment of $L^p$ gradient bounds for heat semigroups.

Abstract

We study Sobolev and BV spaces on local trees which are metric spaces locally isometric to real trees. Such spaces are equipped with a Radon measure satisfying a locally uniform volume growth condition. Using the intrinsic geodesic structure, we define weak gradients and develop from it a coherent theory of Sobolev and BV spaces. We provide two main characterizations: one via Korevaar-Schoen-type energy functionals and another via the heat kernel associated with the natural Dirichlet form. Applications include interpolation results for Besov-Lipschitz spaces, critical exponents computations, and a Nash inequality. In globally tree-like settings we also establish $L^p$ gradient bounds for the heat semigroup.