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A metric approach to sparse domination

José M. Conde Alonso, Francesco Di Plinio, Ioannis Parissis, Manasa N. Vempati

Abstract

We present a general approach to sparse domination based on single-scale $L^p$-improving as a key property. The results are formulated in the setting of metric spaces of homogeneous type and avoid completely the use of dyadic-probabilistic techniques as well as of Christ-Hytönen-Kairema cubes. Among the applications of our general principle, we recover sparse domination of Dini-continuous Calderón-Zygmund kernels on spaces of homogeneous type, we prove a family of sparse bounds for maximal functions associated to convolutions with measures exhibiting Fourier decay, and we deduce sparse estimates for Radon transforms along polynomial submanifolds of $\mathbb R^n$.

A metric approach to sparse domination

Abstract

We present a general approach to sparse domination based on single-scale -improving as a key property. The results are formulated in the setting of metric spaces of homogeneous type and avoid completely the use of dyadic-probabilistic techniques as well as of Christ-Hytönen-Kairema cubes. Among the applications of our general principle, we recover sparse domination of Dini-continuous Calderón-Zygmund kernels on spaces of homogeneous type, we prove a family of sparse bounds for maximal functions associated to convolutions with measures exhibiting Fourier decay, and we deduce sparse estimates for Radon transforms along polynomial submanifolds of .

Paper Structure

This paper contains 27 sections, 23 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. The space of homogeneous type
  4. An operator which is a sum of single scale pieces
  5. Main results
  6. A maximal version
  7. Applications: the $L^p$-improving property revisited
  8. Calderón-Zygmund theory
  9. Calderón-Zygmund operators
  10. Geometric maximal operators
  11. Singular Radon transforms along polynomial manifolds
  12. Homogeneous norms on $\mathbb{R}^n$
  13. $L^p$-improving properties for kernels with Fourier decay
  14. Sparse domination for singular Radon transforms
  15. Whitney covers in geometrically doubling metric spaces and sparse collections
  16. ...and 12 more sections

Key Result

Theorem 1

Let $({\mathbb{X}},{\rm d},|\cdot|)$ be a space of homogeneous type. Let $1\leq p_1\leq p_2 '\leq \infty$, $\omega$ be a Dini modulus of continuity and let $T$ be a linear operator on $({\mathbb{X}},{\rm d},|\cdot|)$ satisfying structural assumption eq:localization. Furthermore, assume: Then, for all $f_1,f_2\in \mathrm{Lip}({\mathbb{X}})$ with compact support and every $\sigma,\tau\in{\mathbb Z}

Theorems & Definitions (44)

  • Remark
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 1
  • Theorem 2
  • Proposition 2.4
  • proof
  • Corollary 2.1
  • Definition 3.1
  • ...and 34 more