Fourier restriction phenomenon in thin sets
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We study the Fourier restriction phenomenon in settings where there is no underlying proper smooth subvariety. We prove an (Lp, L2) restriction theorem in general locally compact abelian groups and apply it in groups such as (Z/pLZ)n, R and locally compact ultrametric fields K. The problem of existence of Salem sets in a locally compact ultrametric field (K, | · |) is also considered. We prove that for every 0 < α < 1 and ǫ > 0 there exist a set E ⊂ K and a measure μ supported on E such that the Hausdorff dimension of E equals α and |bμ(x)| ≤ C|x|−α 2 +ǫ. We also establish the optimal extension of the Hausdorff-Young inequality in the compact ring of integers R of a locally compact ultrametric field K. We shall prove the following: For every 1 ≤ p ≤ 2 there is a Banach function space Fp(R) with σ-order continuous norm such that (i) Lp(R) ( Fp(R) ( L1(R) for every 1 < p < 2. (ii) The Fourier transform F maps Fp(R) to ℓp′ continuously. (iii) Lp(R) is continuously included in Fp(R) and Fp(R) is continuously included in L1(R). (iv) If Z is a Banach function space with the same properties as Fp(R) above, then Z is continuously included in Fp(R). (v) F1(R) = L1(R) and F2(R) = L2(R).