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# Transfer of Fourier Multipliers into Schur Multipliers and Sumsets in a Discrete Group

Published:2011-08-03
Printed: Oct 2011
• Stefan Neuwirth,
Laboratoire de Mathématiques, Université de Franche-Comté, 25000 Besançon, France
• Éric Ricard,
Laboratoire de Mathématiques, Université de Franche-Comté, 25000 Besançon, France
 Format: LaTeX MathJax PDF

## Abstract

We inspect the relationship between relative Fourier multipliers on noncommutative Lebesgue-Orlicz spaces of a discrete group $\varGamma$ and relative Toeplitz-Schur multipliers on Schatten-von-Neumann-Orlicz classes. Four applications are given: lacunary sets, unconditional Schauder bases for the subspace of a Lebesgue space determined by a given spectrum $\varLambda\subseteq\varGamma$, the norm of the Hilbert transform and the Riesz projection on Schatten-von-Neumann classes with exponent a power of 2, and the norm of Toeplitz Schur multipliers on Schatten-von-Neumann classes with exponent less than 1.
 Keywords: Fourier multiplier, Toeplitz Schur multiplier, lacunary set, unconditional approximation property, Hilbert transform, Riesz projection
 MSC Classifications: 47B49 - Transformers, preservers (operators on spaces of operators) 43A22 - Homomorphisms and multipliers of function spaces on groups, semigroups, etc. 43A46 - Special sets (thin sets, Kronecker sets, Helson sets, Ditkin sets, Sidon sets, etc.) 46B28 - Spaces of operators; tensor products; approximation properties [See also 46A32, 46M05, 47L05, 47L20]

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