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Endpoint Fourier restriction and unrectifiability. (English) Zbl 1495.28004

Summary: We show that if a measure of dimension \(s\) on \(\mathbb{R}^d\) admits \((p,q)\) Fourier restriction for some endpoint exponents allowed by its dimension, namely \(q=\frac{s}{d}p'\) for some \(p>1\), then it is either absolutely continuous or \(1\)-purely unrectifiable.

MSC:

28A75 Length, area, volume, other geometric measure theory
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
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