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Multi-scale sparse domination

David Beltran, Joris Roos, Andreas Seeger

TL;DR

This work develops a comprehensive framework for multi-scale sparse domination of linear operators, culminating in a bilinear (p,q') sparse bound for sums T=∑_j T_j under natural scale- and regularity-conditions. The authors extend sparse domination to vector-valued settings, derive necessary conditions, and apply the framework to a broad array of operators, including Fourier multipliers, maximal functions, square functions, and variation-norm operators, with consequences for weighted inequalities. The core contributions include a sharp main theorem, a robust single-scale analysis, a resolution-of-identity tool, and a suite of multiplier theorems that yield new sparse bounds in diverse contexts. The results unify and extend prior sparse domination results beyond Calderón–Zygmund theory, with potential impact on analysis in multiple scales, oscillatory phenomena, and weighted harmonic analysis. Overall, the paper provides a versatile, broadly applicable approach to sparse domination that yields concrete quantitative bounds for a wide class of multi-scale operators.

Abstract

We prove a bilinear form sparse domination theorem that applies to many multi-scale operators beyond Calderón-Zygmund theory, and also establish necessary conditions. Among the applications, we cover large classes of Fourier multipliers, maximal functions, square functions and variation norm operators.

Multi-scale sparse domination

TL;DR

This work develops a comprehensive framework for multi-scale sparse domination of linear operators, culminating in a bilinear (p,q') sparse bound for sums T=∑_j T_j under natural scale- and regularity-conditions. The authors extend sparse domination to vector-valued settings, derive necessary conditions, and apply the framework to a broad array of operators, including Fourier multipliers, maximal functions, square functions, and variation-norm operators, with consequences for weighted inequalities. The core contributions include a sharp main theorem, a robust single-scale analysis, a resolution-of-identity tool, and a suite of multiplier theorems that yield new sparse bounds in diverse contexts. The results unify and extend prior sparse domination results beyond Calderón–Zygmund theory, with potential impact on analysis in multiple scales, oscillatory phenomena, and weighted harmonic analysis. Overall, the paper provides a versatile, broadly applicable approach to sparse domination that yields concrete quantitative bounds for a wide class of multi-scale operators.

Abstract

We prove a bilinear form sparse domination theorem that applies to many multi-scale operators beyond Calderón-Zygmund theory, and also establish necessary conditions. Among the applications, we cover large classes of Fourier multipliers, maximal functions, square functions and variation norm operators.

Paper Structure

This paper contains 54 sections, 52 theorems, 522 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The main result
  3. Necessary conditions
  4. An application to maximal functions
  5. Fourier multipliers
  6. Application to weighted norm inequalities
  7. Structure of the paper and notation
  8. Necessary conditions
  9. The local integrability hypothesis
  10. The reflexivity hypothesis
  11. Single scale sparse domination
  12. A single scale estimate
  13. A resolution of the identity
  14. Single scale regularity
  15. Proof of the main result
  16. ...and 39 more sections

Key Result

Theorem 1.1

Let $1<p\le q<\infty$. Let $\{T_j\}_{j\in{\mathbb {Z}}}$ be a family of operators in $\mathrm{Op}_{B_1,B_2}$ such that Define Then, for all integers $N_1, N_2$ with $N_1\le N_2$,

Figures (3)

  • Figure 1: Example for $\mathscr{L}(\sigma,E)$ (left) and $\mathrm{Sp}[M_E^\sigma]$ (right). It may occur that the closure of $\mathscr{L}(\sigma,E)$ is not a polygonal region, see for example RoosSeeger.
  • Figure 2: Sparse bounds for a general multiplier in $\mathrm{Miy}(a,b)$ (left) and for the oscillatory multipliers $m_{a,b}$ (right) for given $a,b >0$. The condition (ii) in Proposition \ref{['Miyachiprop']} can be relaxed for the specific $m_{a,b}$ (Proposition \ref{['prop:oscmult-msc']}).
  • Figure 3: The region $\mathscr R(\beta,\alpha)$ with $\beta=0.75$, $\alpha=0.9$, $d=3$.

Theorems & Definitions (111)

  • Definition
  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Definition
  • Remark
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 1.7: bernicot-frey-petermichl
  • ...and 101 more