Matrix-Weighted Poincar\'e-Type Inequalities with Applications to Logarithmic Haj{\l}asz--Besov Spaces on Spaces of Homogeneous Type

TL;DR

This paper develops matrix-weighted Poincaré inequalities and introduces logarithmic Besov spaces on spaces of homogeneous type, providing new characterizations independent of reverse doubling conditions.

Contribution

It introduces matrix-weighted Poincaré inequalities and characterizes logarithmic Besov spaces via Haj extasciasharz}asz gradient sequences, independent of reverse doubling conditions.

Findings

Established matrix-weighted Poincaré inequalities on spaces of homogeneous type.

Introduced matrix-weighted logarithmic Besov spaces and their pointwise characterization.

Results are new even for unweighted logarithmic Besov spaces on these spaces.

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 Haj\l asz gradient sequences and establish some related matrix-weighted Poincar\'{e}-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 Haj{\l}asz 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. Hyt\"onen, 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 ${\mathcal {X}}$.