Riesz-Schur transforms

TL;DR

This paper explores nontrigonometric Riesz transforms via Schur multipliers, providing new criteria for Schatten p-boundedness, simplifying proofs of existing theorems, and offering dimension-free estimates with a discrete approach.

Contribution

It introduces a new condition for Schatten p-boundedness of Schur multipliers, refining endpoint criteria and simplifying proofs in harmonic analysis.

Findings

Refined endpoint criterion for Schur multipliers

Simplified discrete approach to Riesz transforms

Dimension-free estimates for trigonometric Riesz transforms

Abstract

We investigate nontrigonometric forms of Riesz transforms in the context of Schur multipliers. This refines Grothendieck-Haagerup's endpoint criterion with a new condition for the Schatten p-boundedness of Schur multipliers and strengthens Potapov/Sukochev's solution of Arazy's conjecture. We recover as well dimension-free estimates for trigonometric Riesz transforms. Our discrete approach is much simpler than previous harmonic analysis and probabilistic approaches. As an application, we find a very simple proof of recent criteria for Schur multipliers of H\"ormander-Mikhlin and Marcinkiewicz type.