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Matrix weights, singular integrals, Jones factorization and Rubio de Francia extrapolation

David Cruz-Uribe

Abstract

In this article we give an overview of the problem of finding sharp constants in matrix weighted norm inequalities for singular integrals, the so-called matrix A2 conjecture. We begin by reviewing the history of the problem in the scalar case, including a sketch of the proof of the scalar A2 conjecture. We then discuss the original, qualitative results for singular integrals with matrix weights and the best known quantitative estimates. We give an overview of new results by the author and Bownik, who developed a theory of harmonic analysis on convex set-valued functions. This led to the proof the Jones factorization theorem and the Rubio de Francia extrapolation theorem for matrix weights, two longstanding problems. Rubio de Francia extrapolation was expected to be a major tool in the proof of the matrix A2 conjecture; however, this conjecture was very recently proved false. We discuss this problem.

Matrix weights, singular integrals, Jones factorization and Rubio de Francia extrapolation

Abstract

In this article we give an overview of the problem of finding sharp constants in matrix weighted norm inequalities for singular integrals, the so-called matrix A2 conjecture. We begin by reviewing the history of the problem in the scalar case, including a sketch of the proof of the scalar A2 conjecture. We then discuss the original, qualitative results for singular integrals with matrix weights and the best known quantitative estimates. We give an overview of new results by the author and Bownik, who developed a theory of harmonic analysis on convex set-valued functions. This led to the proof the Jones factorization theorem and the Rubio de Francia extrapolation theorem for matrix weights, two longstanding problems. Rubio de Francia extrapolation was expected to be a major tool in the proof of the matrix A2 conjecture; however, this conjecture was very recently proved false. We discuss this problem.
Paper Structure (5 sections, 5 theorems, 56 equations)

This paper contains 5 sections, 5 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Background: the scalar theory of weights
  3. Matrix weights and matrix-weighted norm inequalities
  4. Convex set-valued functions and matrix weights
  5. Extrapolation and the failure of the matrix $A_2$ conjecture

Key Result

Theorem 2.1

Given $p_0$, $1\leq p_0<\infty$, suppose that for some operator $T$ and for all $w_0\in A_{p_0}$, the inequality holds. Then for all $p$, $1<p<\infty$, and all $w\in A_p$,

Theorems & Definitions (9)

  • Theorem 2.1
  • Remark 2.2
  • Theorem 4.1
  • Remark 4.2
  • Theorem 4.3
  • Remark 4.4
  • Remark 4.5
  • Theorem 4.6
  • Theorem 4.7