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Orlicz-Sobolev embeddings and heat kernel based Besov classes

Patricia Alonso Ruiz, Fabrice Baudoin

TL;DR

The paper addresses the challenge of variable local-to-global diffusion scaling on metric measure spaces by building Besov-type function spaces anchored in Orlicz norms and heat-kernel-based energies. It develops a Korevaar–Schoen energy framework and establishes Orlicz–Sobolev inequalities, isoperimetric bounds, and a coarea formula, with a strong emphasis on the p=1 BV-like theory. A weak Bakry–Émery condition is shown to imply weak monotonicity and enable semigroup continuity results on Besov classes, unifying BV-geometry and diffusion analysis on fractal-like spaces. The results provide a flexible analytic toolkit for BV, perimeter, and embedding problems in non-smooth, variable-scaling settings, with potential applications to fractal diffusion and geometric analysis on irregular spaces.

Abstract

This paper investigates functional inequalities involving Besov spaces and functions of bounded variation, when the underlying metric measure space displays different local and global structures. Particular focus is put on the $L^1$ theory and its applications to sets of finite perimeter and isoperimetric inequalities, which can now capture such structural differences.

Orlicz-Sobolev embeddings and heat kernel based Besov classes

TL;DR

The paper addresses the challenge of variable local-to-global diffusion scaling on metric measure spaces by building Besov-type function spaces anchored in Orlicz norms and heat-kernel-based energies. It develops a Korevaar–Schoen energy framework and establishes Orlicz–Sobolev inequalities, isoperimetric bounds, and a coarea formula, with a strong emphasis on the p=1 BV-like theory. A weak Bakry–Émery condition is shown to imply weak monotonicity and enable semigroup continuity results on Besov classes, unifying BV-geometry and diffusion analysis on fractal-like spaces. The results provide a flexible analytic toolkit for BV, perimeter, and embedding problems in non-smooth, variable-scaling settings, with potential applications to fractal diffusion and geometric analysis on irregular spaces.

Abstract

This paper investigates functional inequalities involving Besov spaces and functions of bounded variation, when the underlying metric measure space displays different local and global structures. Particular focus is put on the theory and its applications to sets of finite perimeter and isoperimetric inequalities, which can now capture such structural differences.
Paper Structure (15 sections, 24 theorems, 157 equations)

This paper contains 15 sections, 24 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. Generalized heat semigroup based Besov classes
  3. Preliminaries: Orlicz spaces
  4. Generalized heat semigroup based Besov classes
  5. Isoperimetric inequalities
  6. Orlicz-Sobolev inequalities
  7. Korevaar-Schoen-type characterization
  8. Function space characterization
  9. Weak Bakry-Émery condition
  10. Continuity of the semigroup in the Besov class
  11. Pseudo Poincaré inequality
  12. [small]
  13. Weak monotonicity
  14. Coarea formula and sets of finite perimeter
  15. Orlicz-Sobolev embedding

Key Result

Lemma 2.3

Let $\phi$ be a Young function with Young conjugate $\psi$. For any Borel set $E\subset X$ with positive and finite measure,

Theorems & Definitions (55)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Proposition 2.6
  • Lemma 2.7
  • proof
  • Corollary 2.8
  • ...and 45 more