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Trigonometric chaos
and $\mathrm{X}_p$ inequalities, I: Balanced Fourier
truncations over discrete groups
Antonio Ismael Cano-Mármol, José M. Conde-Alonso and Javier Parcet |
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Vol. 17 (2024), No. 7, 2561–2584
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Abstract |
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We investigate -estimates for balanced averages of Fourier truncations in group algebras, in terms of “differential operators” acting on them. Our results extend a fundamental inequality of Naor for the hypercube (with profound consequences in metric geometry) to discrete groups. Different inequalities are established in terms of “directional derivatives” which are constructed via affine representations determined by the Fourier truncations. Our proofs rely on the Banach nature of noncommutative -spaces and dimension-free estimates for noncommutative Riesz transforms. In the particular case of free groups we use an alternative approach based on free Hilbert transforms. |
Keywords
$\mathrm{X}_p$ inequalities, geometry of $L_p$ spaces,
hypercube, discrete groups
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Mathematical Subject Classification
Primary: 42B05, 46L07, 46L53
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Milestones
Received: 30 September 2022
Revised: 18 April 2023
Accepted: 6 June 2023
Published: 21 August 2024
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Authors |
| Antonio Ismael Cano-Mármol | |
| Instituto de Ciencias
Matemáticas Consejo Superior de Investigaciones Científicas Madrid Spain |
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| José M. Conde-Alonso | |
| Universidad Autónoma de Madrid Madrid Spain |
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| Javier Parcet | |
| Instituto de Ciencias
Matemáticas Consejo Superior de Investigaciones Científicas Madrid Spain |
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| © 2024 MSP (Mathematical Sciences Publishers). Distributed under the Creative Commons Attribution License 4.0 (CC BY). |
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