A Marcinkiewicz Testing Criterion for Schur Multipliers

TL;DR

This paper establishes a Marcinkiewicz testing criterion for Schur multipliers on Schatten p-classes, generalizing previous results and providing new decompositions for these operator classes.

Contribution

It introduces a new Marcinkiewicz testing condition for boundedness of Schur multipliers, extending Bourgain's work and offering a novel unconditional decomposition for Schatten p-classes.

Findings

Proved a Marcinkiewicz testing criterion for Schur multipliers.

Generalized Bourgain's results for Toeplitz Schur multipliers.

Derived a new unconditional decomposition for Schatten p-classes.

Abstract

We prove a Marcinkiewicz testing condition for the boundedness of Schur multipliers on the Schatten $p$-classes. This generalizes a previous work of J. Bourgain for Toeplitz type Schur multipliers. As a corollary, we obtain a new unconditional decomposition for the Schatten $p$-classes ($1<p<\infty$).