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Operator-free sparse domination

Andrei K. Lerner, Emiel Lorist, Sheldy Ombrosi

Abstract

We obtain a sparse domination principle for an arbitrary family of functions $f(x,Q)$, where $x\in {\mathbb R}^n$ and $Q$ is a cube in ${\mathbb R}^n$. When applied to operators, this result recovers our recent works. On the other hand, our sparse domination principle can be also applied to non-operator objects. In particular, we show applications to generalized Poincaré-Sobolev inequalities, tent spaces, and general dyadic sums. Moreover, the flexibility of our result allows us to treat operators that are not localizable in the sense of our previous works, as we will demonstrate in an application to vector-valued square functions.

Operator-free sparse domination

Abstract

We obtain a sparse domination principle for an arbitrary family of functions , where and is a cube in . When applied to operators, this result recovers our recent works. On the other hand, our sparse domination principle can be also applied to non-operator objects. In particular, we show applications to generalized Poincaré-Sobolev inequalities, tent spaces, and general dyadic sums. Moreover, the flexibility of our result allows us to treat operators that are not localizable in the sense of our previous works, as we will demonstrate in an application to vector-valued square functions.

Paper Structure

This paper contains 20 sections, 17 theorems, 159 equations.

Table of Contents

  1. Introduction
  2. Main definitions
  3. Dyadic cubes
  4. The lr-condition
  5. Sharp maximal operators
  6. Nonincreasing rearrangements
  7. Main results
  8. A toy domination principle
  9. A pointwise sparse domination principle
  10. A bilinear form sparse domination principle
  11. Sparse domination in spaces of homogeneous type
  12. Previous known results from our sparse domination principles
  13. The local mean oscillation estimate
  14. The lr-sparse domination principle for operators
  15. Generalised Poincaré--Sobolev inequalities
  16. ...and 5 more sections

Key Result

Proposition \oldthetheorem

Let $r\in (0,\infty)$ and let $\{f_Q,f_{P,Q}\}_{Q \in \mathcal{Q}, P \in \mathcal{D}(Q)}$ satisfy the $\ell^r$-condition. Let $Q\in {\mathcal{Q}}$ and let ${\mathcal{F}}\subset {\mathcal{D}}(Q)$ be a contracting family of cubes. Then for a.e. $x\in Q$,

Theorems & Definitions (43)

  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Definition \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • proof
  • Remark \oldthetheorem
  • Theorem \oldthetheorem
  • proof
  • ...and 33 more