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Large sets without Fourier restriction theorems (2001.10016v1)
Published 27 Jan 2020 in math.CA
Abstract: We construct a function that lies in $Lp(\mathbb{R}d)$ for every $p \in (1,\infty]$ and whose Fourier transform has no Lebesgue points in a Cantor set of full Hausdorff dimension. We apply Kova\v{c}'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 {\L}aba and Wang, we hence prove a lack of valid relations between the Hausdorff dimension of a set and the range of possible Fourier restriction exponents for measures supported in the set.