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Extrapolation and Factorization of matrix weights

Marcin Bownik, David Cruz-Uribe

Abstract

In this paper we prove the Jones factorization theorem and the Rubio de Francia extrapolation theorem for matrix $\mathcal A_p$ weights. These results answer longstanding open questions in the study of matrix weights. The proof requires the development of the theory of convex-set valued functions and measurable seminorm functions. In particular, we define a convex-set valued version of the Hardy Littlewood maximal operator and construct an appropriate generalization of the Rubio de Francia iteration algorithm, which is central to the proof of both results in the scalar case.

Extrapolation and Factorization of matrix weights

Abstract

In this paper we prove the Jones factorization theorem and the Rubio de Francia extrapolation theorem for matrix weights. These results answer longstanding open questions in the study of matrix weights. The proof requires the development of the theory of convex-set valued functions and measurable seminorm functions. In particular, we define a convex-set valued version of the Hardy Littlewood maximal operator and construct an appropriate generalization of the Rubio de Francia iteration algorithm, which is central to the proof of both results in the scalar case.
Paper Structure (12 sections, 78 theorems, 375 equations)

This paper contains 12 sections, 78 theorems, 375 equations.

Table of Contents

  1. Introduction
  2. Convex sets and seminorms
  3. Convex-set valued functions
  4. Seminorm functions
  5. The maximal operator on convex-set valued functions
  6. Matrix $\mathcal{A}_p$ weights and weighted norm inequalities
  7. Convex-set valued $\mathcal{A}_1$ and the Rubio de Francia iteration algorithm
  8. Factorization of matrix weights
  9. Factorization
  10. Reverse factorization
  11. Extrapolation of matrix weights
  12. An application of Rubio de Francia extrapolation

Key Result

Theorem 1.1

Given a weight $w$ and $1<p<\infty$, $w\in A_p$ if and only if there exist weights $w_0,\,w_1\in A_1$ such that $w=w_0w_1^{1-p}$.

Theorems & Definitions (161)

  • Theorem 1.1: Jones Factorization Theorem
  • Theorem 1.2: Rubio de Francia Extrapolation
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Remark 1.9
  • Definition 2.1
  • ...and 151 more