Endpoint Fourier restriction and unrectifiability

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 \(q=\tfrac{s}{d}p'\) for some \(p>1\), then it is either absolutely continuous or \(1\)-purely unrectifiable.

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Bibliographic Details
Published in:arXiv.org 2021-04
Main Authors: Giacomo Del Nin, Merlo, Andrea
Format: Article
Language:English
Online Access:Get full text
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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 \(q=\tfrac{s}{d}p'\) for some \(p>1\), then it is either absolutely continuous or \(1\)-purely unrectifiable.
ISSN:2331-8422