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Matrix-Weighted Poincaré-Type Inequalities with Applications to Logarithmic Hajłasz--Besov Spaces on Spaces of Homogeneous Type

Ziwei Li, Dachun Yang, Wen Yuan

Abstract

Let ${\mathcal {X}}$ be a space of homogeneous type. In this article, based on the reducing operators of matrix $A_p$-weights, the authors introduce the vector-valued Hajłasz gradient sequences and establish some related matrix-weighted Poincaré-type inequalities on ${\mathcal {X}}$. As an application, the authors introduce the matrix-weighted logarithmic Besov spaces on ${\mathcal {X}}$ and establish their pointwise characterization via Hajłasz gradient sequences. The novelty of this article lies in that, by means of both the $A_p$ dimension and its properties of matrix $A_p$-weights and the wavelet reproducing formula with exponential decay of P. Auscher and T. Hytönen, all the main results get rid of the dependence on the reverse doubling conditions of both weights and ${\mathcal {X}}$ under consideration and these results are also completely new even for unweighted logarithmic Besov spaces on ${\mathcal {X}}$.

Matrix-Weighted Poincaré-Type Inequalities with Applications to Logarithmic Hajłasz--Besov Spaces on Spaces of Homogeneous Type

Abstract

Let be a space of homogeneous type. In this article, based on the reducing operators of matrix -weights, the authors introduce the vector-valued Hajłasz gradient sequences and establish some related matrix-weighted Poincaré-type inequalities on . As an application, the authors introduce the matrix-weighted logarithmic Besov spaces on and establish their pointwise characterization via Hajłasz gradient sequences. The novelty of this article lies in that, by means of both the dimension and its properties of matrix -weights and the wavelet reproducing formula with exponential decay of P. Auscher and T. Hytönen, all the main results get rid of the dependence on the reverse doubling conditions of both weights and under consideration and these results are also completely new even for unweighted logarithmic Besov spaces on .
Paper Structure (8 sections, 26 theorems, 156 equations)

This paper contains 8 sections, 26 theorems, 156 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notation and notions
  4. Matrix $A_p$-Weights on Spaces of Homogeneous Type
  5. Matrix-Weighted Poincaré-Type Inequalities
  6. Characterization of Logarithmic Matrix-Weighted Besov Spaces via Hajłasz Gradient Sequences
  7. Grand Maximal Function Characterization of Logarithmic Matrix-Weighted Besov Spaces
  8. Pointwise Characterizations of Matrix-Weighted Logarithmic Besov Spaces

Key Result

Lemma 2.6

Let $p\in(0,\infty)$, $W\in A_p(\mathcal{X},\mathbb{C}^m)$, $\mathcal{P}$ be a given family of measurable subsets with finite positive measure of $\mathcal{X}$ and $\mathcal{RS}_{\mathcal{P}}^{W,p}$ a family of reducing operators of order $p$ for $W$ with respect to $\mathcal{P}$. Then, for any boun and, especially, when $p\in(1,\infty)$, where the positive equivalence constants are all independe

Theorems & Definitions (64)

  • Definition 2.1
  • Remark 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Lemma 2.6
  • Definition 2.7
  • Lemma 2.8
  • Remark 2.9
  • Lemma 2.10
  • ...and 54 more