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Convex body domination for rough singular integrals

Aapo Laukkarinen

Abstract

Convex body domination is a technique, where operators acting on vector-valued functions are estimated via certain convex body averages of the input functions. This domination lets one deduce various matrix weighted bounds for these operators and their commutators. In this paper, we extend the sparse domination results for rough singular integrals due to Conde-Alonso, Culiuc, Di Plinio and Ou to the convex body setting. In particular, our methods apply to homogeneous rough singular integrals with unbounded angular part. We also note that convex body domination implies new two weight commutator bounds even in the scalar case.

Convex body domination for rough singular integrals

Abstract

Convex body domination is a technique, where operators acting on vector-valued functions are estimated via certain convex body averages of the input functions. This domination lets one deduce various matrix weighted bounds for these operators and their commutators. In this paper, we extend the sparse domination results for rough singular integrals due to Conde-Alonso, Culiuc, Di Plinio and Ou to the convex body setting. In particular, our methods apply to homogeneous rough singular integrals with unbounded angular part. We also note that convex body domination implies new two weight commutator bounds even in the scalar case.

Paper Structure

This paper contains 8 sections, 20 theorems, 130 equations.

Table of Contents

  1. Introduction
  2. Sesquilinear forms and stopping collections
  3. Complex convex bodies
  4. An abstract convex body domination principle
  5. Rough homogeneous singular integrals
  6. Matrix weighted inequalities
  7. Sparse domination of commutators and Bloom-type bounds
  8. Bochner-Riesz means at the critical index

Key Result

Lemma 2.2

Assume that $K$ satisfies supportCond and KernelCond with $1<q\leq\infty$. If $b\in\mathcal{X}_1(\mathcal{Q})$ with $\operatorname{supp} b\subset Q$, then where If $h\in \mathcal{Y}_{q'}(\mathcal{Q})$ with $\operatorname{supp} h\subset Q$, and $b\in \mathcal{X}_{q'}(\mathcal{Q})$, then where $\Lambda^*$ is like $\Lambda$, but with $\overline{K(y,x)}$ in place of $K(x,y)$, the function $b^\text{

Theorems & Definitions (32)

  • Remark 2.1
  • Lemma 2.2
  • proof
  • Lemma 3.1
  • Lemma 4.1
  • Proposition 4.2
  • proof
  • Theorem 4.3
  • proof
  • Lemma 4.4
  • ...and 22 more