Operator-free sparse domination

TL;DR

This paper introduces a versatile sparse domination principle applicable to a wide range of functions and objects, extending previous operator-focused results to non-operator contexts like inequalities and function spaces.

Contribution

It presents a novel, operator-free sparse domination framework that applies to various non-operator mathematical objects and broadens the scope of sparse domination techniques.

Findings

Unified sparse domination principle for functions and operators

Applications to Poincaré-Sobolev inequalities and tent spaces

Extension to non-localizable operators like vector-valued square functions

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 Poincar\'e-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.