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

Fabrice Baudoin, Li Chen, Meng Yang

TL;DR

This work develops a robust analysis of Sobolev and BV spaces on local trees, spaces that are locally 1-dimensional yet may have complex global geometry. It provides two parallel, deep characterizations: Korevaar-Schoen type energies $E_{p,\Psi_p}(f,r)$ and heat-kernel based norms tied to the natural Dirichlet form, establishing equivalences with $W^{1,p}(X,m)$ for $p>1$ and $BV(X,m)$ when $p=1$. The KS framework is complemented by a detailed heat kernel theory, including sub-Gaussian bounds, off-diagonal estimates, Besov-Lipschitz interpolation results, and Nash inequalities, all adapted to the local-tree geometry through the volume growth function $\Phi$ and the derived scale $\Psi_p$. Applications include real interpolation of Besov-Lipschitz spaces, critical exponent computations, Nash inequalities, and, in globally tree-like settings, $L^p$ gradient bounds for the heat semigroup. Together, these results extend Sobolev/BV analysis to non-smooth, tree-like metric measure spaces, linking geometric structure to analytic function spaces and semigroup regularity.

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.

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

TL;DR

This work develops a robust analysis of Sobolev and BV spaces on local trees, spaces that are locally 1-dimensional yet may have complex global geometry. It provides two parallel, deep characterizations: Korevaar-Schoen type energies and heat-kernel based norms tied to the natural Dirichlet form, establishing equivalences with for and when . The KS framework is complemented by a detailed heat kernel theory, including sub-Gaussian bounds, off-diagonal estimates, Besov-Lipschitz interpolation results, and Nash inequalities, all adapted to the local-tree geometry through the volume growth function and the derived scale . Applications include real interpolation of Besov-Lipschitz spaces, critical exponent computations, Nash inequalities, and, in globally tree-like settings, gradient bounds for the heat semigroup. Together, these results extend Sobolev/BV analysis to non-smooth, tree-like metric measure spaces, linking geometric structure to analytic function spaces and semigroup regularity.

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 gradient bounds for the heat semigroup.
Paper Structure (18 sections, 34 theorems, 160 equations, 5 figures)

This paper contains 18 sections, 34 theorems, 160 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Sobolev spaces on local trees
  3. Local trees
  4. Weak gradients
  5. Sobolev, BV spaces and $p$-energy forms
  6. Korevaar-Schoen type characterizations of the Sobolev and BV spaces and applications
  7. Controlled partitions of unity
  8. Regularity of the $p$-energy forms
  9. Korevaar-Schoen characterization of the Sobolev and BV spaces
  10. Critical exponents and real interpolation of the Besov-Lipschitz spaces
  11. Nash inequality
  12. Heat kernel characterization of the Sobolev and BV spaces
  13. Heat kernel estimates
  14. Heat kernel characterization of the Sobolev and BV spaces
  15. Local heat kernel gradient estimates
  16. ...and 3 more sections

Key Result

Lemma 2.9

Let $f \in AC(X)$ and $x \in X$, then for every $u,v \in B(x,\iota(x))$

Figures (5)

  • Figure 1: The Sierpiński carpet cable system
  • Figure 2: Tripod defining $c(u,v,w)$
  • Figure 3: Vicsek set
  • Figure 4: Example of a local tree: Blow up of the Vicsek set by another fractal
  • Figure 5: Example of a local tree

Theorems & Definitions (74)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4: Cable systems
  • Example 2.5: Vicsek set
  • Example 2.6
  • Definition 2.7
  • Definition 2.8
  • Lemma 2.9
  • proof
  • ...and 64 more