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Large sets without Fourier restriction theorems.
- Source :
- Transactions of the American Mathematical Society; Oct2022, Vol. 375 Issue 10, p6983-7000, 18p
- Publication Year :
- 2022
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- FRACTAL dimensions
FOURIER transforms
MEASURE theory
POINT set theory
CANTOR sets
Subjects
Details
- Language :
- English
- ISSN :
- 00029947
- Volume :
- 375
- Issue :
- 10
- Database :
- Complementary Index
- Journal :
- Transactions of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 159443027
- Full Text :
- https://doi.org/10.1090/tran/8714