Riesz-Schur transforms
Adrian Gonz\'alez-P\'erez, Javier Parcet, Jorge P\'erez Garc\'ia and, \'Eric Ricard

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.
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Taxonomy
TopicsMathematical functions and polynomials · Fractional Differential Equations Solutions
