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Cancellative sparse domination

José M. Conde Alonso, Emiel Lorist, Guillermo Rey

Abstract

We present a general sparse domination principle which respects the cancellative structure of the functions under study. We obtain sparse domination results in general measure spaces, including general martingale settings in one and two parameters, and in the Euclidean setting. In the one-parameter martingale setting, we obtain a sparse characterization of the $H^1$ norm. The proofs make critical use of precise level-set estimates for generalized versions of medians. Our results imply new, quantitatively sharp, weighted results for martingales and Calderón-Zygmund operators acting on $H^p$ spaces.

Cancellative sparse domination

Abstract

We present a general sparse domination principle which respects the cancellative structure of the functions under study. We obtain sparse domination results in general measure spaces, including general martingale settings in one and two parameters, and in the Euclidean setting. In the one-parameter martingale setting, we obtain a sparse characterization of the norm. The proofs make critical use of precise level-set estimates for generalized versions of medians. Our results imply new, quantitatively sharp, weighted results for martingales and Calderón-Zygmund operators acting on spaces.
Paper Structure (9 sections, 22 theorems, 220 equations, 1 figure)

This paper contains 9 sections, 22 theorems, 220 equations, 1 figure.

Table of Contents

  1. General martingales - single-scale operators
  2. Regular martingales - multiscale operators and dyadic harmonic analysis
  3. Martingale transforms and square functions
  4. Dyadic harmonic analysis: Haar shifts
  5. Failure of usual maximal function domination
  6. Calderón--Zygmund operators - Euclidean harmonic analysis
  7. Weighted norm estimates
  8. Muckenhoupt weighted Hardy spaces
  9. Weighted norm estimates for Calderón--Zygmund operators

Key Result

Theorem 1

Let $T$ be a martingale transform, a Haar shift operator, or $\mathcal{M}_\mathscr{D}$. There exists $0<r<1$ such that for each $f\in L_c^\infty(\mathbb{R}^d)$, there is a sparse $\mathcal{S} \subset \mathscr{D}$ so that

Figures (1)

  • Figure 1: Construction of $T_4f$.

Theorems & Definitions (49)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1.1
  • proof
  • Definition 1.2
  • Lemma 1.3
  • proof
  • Proposition 1.4
  • ...and 39 more