Salem sets in local fields, the Fourier restriction phenomenon and the Hausdorff-Young inequality
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 \(\ell_{p'} ({\mathbb Z})\) for every Borel subset \(A\) of \({\mathbb T}\). In the paper under review, the author proves these results for ultrametric local fields. A local field is a topological field \(K\) for which the additive and the multiplicative groups are locally compact abelian groups. Such a field can be endowed with an absolute value (using the uniqueness, up to constants, of the Haar measure) which fits nicely into the topology of \(K\). The closed unit ball centered at \(0\) for this absolute value is a maximal subring \(R\) of \(K\) and is called the ring of integers of \(K\). The author proves, assuming that the absolute value of \(K\) is ultrametric, that there exist Salem subsets of arbitrary Hausdorff dimension \(\alpha\) (\(0 < \alpha < 1\)) in the ring of integers \(R\). His proof follows the probabilistic approach of Salem and needs arithmetical and combinatorial lemmas. He then proves the restriction Fourier phenomenon and uses it to obtain the optimal extension of the Hausdorff-Young inequality in \(R\).
scientific article; zbMATH DE number 5731599
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