Salem sets in local fields, the Fourier restriction phenomenon and the Hausdorff-Young inequality
type: Item
instance of: scholarly article
title: Salem sets in local fields, the Fourier restriction phenomenon and the Hausdorff-Young inequality
zbMATH Open document ID: 1197.43005
author: Christos Papadimitropoulos
published in: Journal of Functional Analysis
publication date: 2010-07-07T00:00:00Z
review text: A compact subset \(E\) of \({\mathbb R}\) is called a Salem set if its Hausdorff dimension \(\alpha\) (\(0 < \alpha < 1\)) is equal to its Fourier dimension, i.e., \(E\) carries, for every \(\beta < \alpha\), positive measures \(\mu_\beta\) whose Fourier transforms decay more quickly than \(|t|^{-\beta/2}\). Such sets were first constructed by \textit{R. Salem} [Ark. Mat. 1, 353--365 (1951; Zbl 0054.03001)] (see also \textit{J.-P. Kahane} [C. R. Acad. Sci., Paris, Sér. A 263, 613--615 (1966; Zbl 0158.35703)]). Ten years ago \textit{G. Mockenhaupt} [Geom. Funct. Anal. 10, No.~6, 1579--1587 (2000; Zbl 0974.42013)] showed that for such a set \(E\), one has the Fourier restriction phenomenon, for \(p > 1\) small enough (depending on the Hausdorff dimension \(\alpha\)): \[ \bigg( \int_E |\hat f (x)|^2\,d\mu_\beta \bigg)^{1/2} \leq C\, \| f \|_{L^p ({\mathbb R})}, \quad \forall f \in L^p ({\mathbb R}). \] This is the \(1\)-dimensional version of a classical result of Stein in dimensions \(d \geq 2\) (where \(E\) is the unit sphere \({\mathbb S}^{d - 1}\)). Later, \textit{G. Mockenhaupt} and \textit{W. J. Ricker} [J. Reine Angew. Math. 620, 195--211 (2008; Zbl 1166.46015)] used this result to show that, for \(1 \leq p \leq 2\), there is a Banach function space \({\mathbb F}^p ({\mathbb T})\), strictly bigger than \(L^p ({\mathbb T})\) for \(1 < p < 2\), and strictly contained in \(L^1 ({\mathbb T})\), for which the Hausdorff-Young inequality \(\| \hat f \|_{\ell_{p'} ({\mathbb Z})} \leq \| f \|_{{\mathbb F}^p ({\mathbb T})}\) holds for every \(f \in {\mathbb F}^p ({\mathbb T})\) (\(p'\) the conjugate number of \(p\)). Moreover, this space is the maximal one, in some natural sense, to which the Hausdorff-Young inequality extends. This space \({\mathbb F}^p ({\mathbb T})\) has several descriptions; the most striking being that \(f \in {\mathbb F}^p ({\mathbb T})\) if and only if \(f \in L^1 ({\mathbb T})\) and the Fourier transform of \(f.{\mathbf 1}_A\) is in \(\ell_{p'} ({…
reviewed by: Daniel Li
Mathematics Subject Classification ID: 43A46
Mathematics Subject Classification ID: 11S99
Mathematics Subject Classification ID: 12J10
Mathematics Subject Classification ID: 12J25
Mathematics Subject Classification ID: 43A25
zbMATH DE Number: 5731599
zbMATH Keywords: Fourier dimension
zbMATH Keywords: Fourier transform
zbMATH Keywords: Hausdorff dimension
zbMATH Keywords: Hausdorff-Young inequality
zbMATH Keywords: local field
zbMATH Keywords: Salem set
zbMATH Keywords: ultrametric local field
zbMATH Keywords: vector measure
MaRDI profile type: Publication
full work available at URL: j.jfa.2010.03.017
OpenAlex ID: W2085528827
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DOI: 10.1016/J.JFA.2010.03.017
description: scientific article; zbMATH DE number 5731599