Some remarks on convex body domination

TL;DR

This paper develops an abstract framework for convex body domination, extending sparse domination techniques to Banach space-valued functions and matrix-weighted inequalities, with applications to generalized commutators.

Contribution

It introduces a novel abstract framework for convex body domination applicable to Banach space-valued functions and matrix weights, expanding the scope of sparse domination methods.

Findings

Established a new framework for convex body domination.

Derived matrix-weighted norm inequalities in Banach spaces.

Provided new examples of bounded operators involving commutators.

Abstract

Convex body domination is an important elaboration of the technique of sparse domination that has seen significant development and applications over the past ten years. In this paper, we present an abstract framework for convex body domination, which also applies to Banach space -valued functions, and yields matrix-weighted norm inequalities in this setting. We explore applications to "generalised commutators", obtaining new examples of bounded operators among linear combinations of compositions of pointwise multipliers and a singular integral operator.