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Matrix $A_p$-weights relative to a pseudo-metric

Morten Nielsen

TL;DR

This work develops a comprehensive theory of matrix weights relative to anisotropic geometric settings defined by pseudo-metrics and one-parameter dilation groups. It proves that such matrix $A_p$-weights satisfy doubling and reverse Hölder properties and demonstrates an invariance under anisotropic affine transformations, enabling boundedness results for band-limited Fourier multipliers and sampling in matrix-weighted spaces. The paper then extends these analytic tools to construct and analyze anisotropic matrix-weighted Besov spaces, including discrete characterizations via tight frames and stability results, thereby providing a robust framework for anisotropic, matrix-weighted harmonic analysis with potential applications to related function spaces and operators. The approach unifies geometric, probabilistic, and analytic techniques to handle matrix-valued weights in non-Euclidean geometries, offering new invariance properties and practical discretization tools. These contributions advance the understanding of weighted norm inequalities in anisotropic contexts and facilitate applications to Besov-type spaces and related operator theory.

Abstract

Matrix weights satisfying a Muckenhoupt $A_p$-condition relative to a family of anisotropic balls in $\mathbb{R}^d$ defined by a pseudo-metric are studied. It is shown that such matrix weights satisfy a doubling condition and a reverse Hölder inequality. In the special case, where the pseudo-metric is homogeneous with respect to a one-parameter dilation group, the corresponding Muckenhoupt class is shows to satisfy an invariance property under composition with affine transformations generated by the dilation group. A general sampling theorem is derived for the matrix-weighted space $L^p(W)$ for Muckenhoupt $A_p$ weights $W$ along with a corresponding multiplier result for $L^p(W)$. An application of the results to the study of anisotropic matrix-weighed Besov spaces is considered.

Matrix $A_p$-weights relative to a pseudo-metric

TL;DR

This work develops a comprehensive theory of matrix weights relative to anisotropic geometric settings defined by pseudo-metrics and one-parameter dilation groups. It proves that such matrix -weights satisfy doubling and reverse Hölder properties and demonstrates an invariance under anisotropic affine transformations, enabling boundedness results for band-limited Fourier multipliers and sampling in matrix-weighted spaces. The paper then extends these analytic tools to construct and analyze anisotropic matrix-weighted Besov spaces, including discrete characterizations via tight frames and stability results, thereby providing a robust framework for anisotropic, matrix-weighted harmonic analysis with potential applications to related function spaces and operators. The approach unifies geometric, probabilistic, and analytic techniques to handle matrix-valued weights in non-Euclidean geometries, offering new invariance properties and practical discretization tools. These contributions advance the understanding of weighted norm inequalities in anisotropic contexts and facilitate applications to Besov-type spaces and related operator theory.

Abstract

Matrix weights satisfying a Muckenhoupt -condition relative to a family of anisotropic balls in defined by a pseudo-metric are studied. It is shown that such matrix weights satisfy a doubling condition and a reverse Hölder inequality. In the special case, where the pseudo-metric is homogeneous with respect to a one-parameter dilation group, the corresponding Muckenhoupt class is shows to satisfy an invariance property under composition with affine transformations generated by the dilation group. A general sampling theorem is derived for the matrix-weighted space for Muckenhoupt weights along with a corresponding multiplier result for . An application of the results to the study of anisotropic matrix-weighed Besov spaces is considered.

Paper Structure

This paper contains 8 sections, 17 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Structured bases and corresponding Muckenhoupt conditions
  3. Weights relative to bases generated by pseudo-metrics
  4. A homogeneous structure on ${\mathbb R}^d$
  5. Fourier multipliers and sampling
  6. Anisotropic matrix-weighted Besov spaces
  7. Tight frames for matrix-weighted Besov spaces
  8. Completeness of Anisotropic matrix-weighted Besov spaces

Key Result

Proposition 2.4

Consider a matrix-valued function $W\colon {\mathbb{R}^d}\rightarrow {\mathbb C}^{N\times N}$, which is measurable, locally integrable, and strictly positive definite a.e. For $1<p<\infty$, we have $W\in \mathbf{A}_p(\mathcal{B})$ if and only if the averaging operators are uniformly bounded on $L^p(W)$ for $E\in\mathcal{B}$. Moreover, the constant $[W]_{{\mathbf{A}_p(\mathcal{B})}}$ is equivalent

Theorems & Definitions (47)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Lemma 2.5
  • proof
  • Corollary 2.6
  • proof
  • Lemma 3.1
  • proof
  • ...and 37 more