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Convex body domination for the commutator of vector valued operators with matrix multi-symbol

Joshua Isralowitz, Israel P. Rivera-Ríos, Francisco Sáez-Rivas

Abstract

We provide convex body domination results for the generalized vector-valued commutator of those operators that admit specific forms of convex body domination themselves. We also prove some strong type estimates and other consequences of these results, and we study the BMO spaces that appear naturally in this context.

Convex body domination for the commutator of vector valued operators with matrix multi-symbol

Abstract

We provide convex body domination results for the generalized vector-valued commutator of those operators that admit specific forms of convex body domination themselves. We also prove some strong type estimates and other consequences of these results, and we study the BMO spaces that appear naturally in this context.
Paper Structure (15 sections, 37 theorems, 227 equations)

This paper contains 15 sections, 37 theorems, 227 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Convex Body Domination for higher order commutators
  4. Quantitative weighted estimates
  5. Definitions and preliminaries
  6. Matrix weights
  7. Combinatorics
  8. Key Lemma relating combinatorics, symbols and operators
  9. $BMO$ spaces
  10. A particular case of interest
  11. Proof of the main results
  12. Proof of Theorems \ref{['th:tocho']} and \ref{['th:tocho3']}
  13. Proof of the strong type estimates
  14. An estimate in terms of bumped weighted $BMO$ spaces
  15. Applications of Theorem \ref{['th:tocho']}

Key Result

Lemma 1

Consider $r\in [1,\infty)$, $Q\subset \mathbb{R}^d$ with finite measure, $\vec{f}\in L^r(Q,\mathbb{F}^n)$ and $\vec{g}:\mathbb{R}^d\rightarrow \mathbb{F}^n$ such that $g(x)\in \langle \langle \vec{f}\rangle\rangle_{r,Q}$ a.e. on $\mathbb{R}^d$. Then there exists $K:\mathbb{R}^d\times Q\rightarrow \m

Theorems & Definitions (67)

  • Lemma 1
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Remark 1
  • Theorem 6
  • ...and 57 more