Large sets without Fourier restriction theorems

Bilz, Constantin

doi:10.1090/tran/8714

Large sets without Fourier restriction theorems
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by Constantin Bilz PDF

Trans. Amer. Math. Soc. 375 (2022), 6983-7000 Request permission

Abstract:

We construct a function that lies in $L^p(\mathbb {R}^d)$ for every $p \in (1,\infty ]$ and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kovač’s maximal restriction principle to show that the same full-dimensional set is avoided by any Borel measure satisfying a nontrivial Fourier restriction theorem. As a consequence of a near-optimal fractal restriction theorem of Łaba and Wang, we hence prove that there are no previously unknown relations between the Hausdorff dimension of a set and the range of possible Fourier restriction exponents for measures supported in the set.

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