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Off-diagonal matrix extrapolation for Muckenhoupt bases

David Cruz-Uribe, Fatih Şirin

Abstract

In this paper we extend the theory of Rubio de Francia extrapolation for matrix weights, recently introduced by Bownik and the first author, to off-diagonal extrapolation. We also show that the theory of matrix weighted extrapolation can be extended to matrix $\mathcal{A}_p$ classes defined with respect to a general basis, provided that a version of the Christ-Goldberg maximal operator is assumed to be bounded. Finally, we extend a recent result by Vuorinen and show that all of the multiparameter bases have this property.

Off-diagonal matrix extrapolation for Muckenhoupt bases

Abstract

In this paper we extend the theory of Rubio de Francia extrapolation for matrix weights, recently introduced by Bownik and the first author, to off-diagonal extrapolation. We also show that the theory of matrix weighted extrapolation can be extended to matrix classes defined with respect to a general basis, provided that a version of the Christ-Goldberg maximal operator is assumed to be bounded. Finally, we extend a recent result by Vuorinen and show that all of the multiparameter bases have this property.

Paper Structure

This paper contains 6 sections, 26 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Matrix Muckenhoupt weights
  3. Convex-set valued operators defined on bases
  4. Proof of Theorem \ref{['thm:off-diag-extrapol']}
  5. The weak matrix Muckenhoupt property
  6. Multiparameter $\mathcal{A}_{p}$ matrix weights

Key Result

Theorem 1.1

Let $T$ be a sublinear operator. Suppose that for some $p_0$, $1 \leq p_0\leq\infty$, there exists an increasing function $N_{p_0}$ such that for every $W_0\in \mathcal{A}_{p_0}$, Then for all $p$, $1<p<\infty$, and for all $W\in \mathcal{A}_p$, where

Theorems & Definitions (55)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Remark 1.4
  • Definition 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 45 more