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Generalised Hajłasz-Besov spaces on $RD$-spaces

Joaquim Martin, Walter A. Ortiz

TL;DR

The work develops a broad, general framework of Besov-type spaces on RD-spaces by coupling a rearrangement-invariant base space $X$ with a parameter space $E$ and a slowly varying modulation $b$, defining the Generalised Hajłasz-Besov spaces $\dot{\mathcal{B}}_{X,E}^{\theta,b}$. It then establishes a comprehensive set of embeddings, connects these spaces to real interpolation theory, and proves Sobolev-type embedding theorems and essential continuity results within this unified approach. The main technical contributions include embedding hierarchies under convexifications and $q$-powers, an interpolation-space characterization $\text{\r{B}}_{X^{(p)},E}^{\theta,b}=(X^{(p)},\text{M}^{1,X^{(p)}})_{\theta,b,E}$, and sharp Sobolev-type estimates that depend on the Boyd indices and the RD-space indices $(k,n)$. Collectively, these results extend classical Besov and Hajłasz-Sobolev theory to a broad metric-measure setting, with potential applications to PDEs and analysis on fractal-like spaces where the geometry is encoded by $X$, $E$, and $b$.

Abstract

An $RD$ space is a doubling measure metric space $Ω$ with the additional property that it has a reverse doubling property. In this paper we introduce a new class of Hajłasz-Besov spaces on $Ω$ and extend several results from classical theory, such as embeddings and Sobolev-type embeddings.

Generalised Hajłasz-Besov spaces on $RD$-spaces

TL;DR

The work develops a broad, general framework of Besov-type spaces on RD-spaces by coupling a rearrangement-invariant base space with a parameter space and a slowly varying modulation , defining the Generalised Hajłasz-Besov spaces . It then establishes a comprehensive set of embeddings, connects these spaces to real interpolation theory, and proves Sobolev-type embedding theorems and essential continuity results within this unified approach. The main technical contributions include embedding hierarchies under convexifications and -powers, an interpolation-space characterization , and sharp Sobolev-type estimates that depend on the Boyd indices and the RD-space indices . Collectively, these results extend classical Besov and Hajłasz-Sobolev theory to a broad metric-measure setting, with potential applications to PDEs and analysis on fractal-like spaces where the geometry is encoded by , , and .

Abstract

An space is a doubling measure metric space with the additional property that it has a reverse doubling property. In this paper we introduce a new class of Hajłasz-Besov spaces on and extend several results from classical theory, such as embeddings and Sobolev-type embeddings.

Paper Structure

This paper contains 11 sections, 13 theorems, 161 equations.

Table of Contents

  1. INTRODUCTION
  2. Notation and preliminaries
  3. Rearrangements of functions and rearrangement invariant spaces.
  4. Parameters spaces, extension indices and slowly varying functions
  5. Parameters Spaces
  6. Slowly varying functions
  7. Extension indices
  8. Embeddings of Generalised Hajłasz-Besov spaces
  9. Generalized Hajłasz-Besov spaces as interpolation spaces
  10. Sobolev embedding theorems for Hajłasz-Besov spaces.
  11. Appendix

Key Result

Lemma 3

Let $E$ be a parameter and $b\in SV$. For all $\alpha>0$ we get

Theorems & Definitions (34)

  • Definition 1
  • Example 2
  • Lemma 3
  • Proposition 4
  • Proposition 5
  • proof
  • Proposition 6
  • proof
  • Lemma 7
  • Theorem 8
  • ...and 24 more