## Fourier restriction phenomenon in thin sets

##### Abstract

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).