Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Matrix weighted estimates on spaces of homogeneous type

Guido Claro, Pamela Muller, Luis Nowak, Alejandra Perini, Israel P. Rivera-Ríos

Abstract

In this paper matrix quantitative weighted estimates on spaces of homogeneous type, such as endpoint estimates, strong type estimates are provided. To that end we extend some earlier results on convex body domination due to Nazarov, Petermichl, Treil and Volberg to this setting. We also provide a $T(1)$ alike convex body domination result analogous to the one provided by Lerner and Ombrosi, and an application to vector valued extensions of Petermichl operators.

Matrix weighted estimates on spaces of homogeneous type

Abstract

In this paper matrix quantitative weighted estimates on spaces of homogeneous type, such as endpoint estimates, strong type estimates are provided. To that end we extend some earlier results on convex body domination due to Nazarov, Petermichl, Treil and Volberg to this setting. We also provide a alike convex body domination result analogous to the one provided by Lerner and Ombrosi, and an application to vector valued extensions of Petermichl operators.

Paper Structure

This paper contains 19 sections, 22 theorems, 246 equations.

Table of Contents

  1. Introduction and Main Results
  2. Weighted estimates
  3. Sparse T(1)-type theorem.
  4. Preliminaries
  5. Dyadic structures
  6. The $A_{p}$ condition
  7. Reducing matrices
  8. Reverse Hölder inequality
  9. Weighted estimates
  10. Proof of Theorem \ref{['thm:StrongMax']}
  11. Proof of Theorem \ref{['Thm:StrongTypeLrH']}
  12. Endpoint estimates
  13. Proof of the $T(1)$-like sparse result
  14. Lemmatta
  15. Proof of Theorem \ref{['thm:sparseT1']}
  16. ...and 4 more sections

Key Result

Theorem 1

Let $(X,d,\mu)$ be a space of homogeneous type and $\mathcal{D}$ a dyadic system with parameters $c_{0}$, $C_{0}$ and $\delta$. Let us fix $\alpha\geq\frac{3c_{d}^{2}}{\delta}$ and let $\vec{f}:X\rightarrow\mathbb{R}^{n}$ be a boundedly supported function such that $|\vec{f}|\in L^{s}(X)$. Let $1\le and Then there exists a $\frac{1}{2}$-sparse family $\mathcal{S}\subset\mathcal{D}$ such that In

Theorems & Definitions (37)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Proposition 1
  • Lemma 2
  • Definition 1
  • Definition 2
  • Proposition 2
  • Lemma 3
  • ...and 27 more