Improved Weighted Restriction Estimates in R3

Suppose 0 < α ≤ n , H : R n → [ 0 , 1 ] is a Lebesgue measurable function, and A α ( H ) is the infimum of all numbers C for which the inequality ∫ B H ( x ) d x ≤ C R α holds for all balls B ⊂ R n of radius R ≥ 1 . After Guth introduced polynomial partitioning to Fourier restriction theory, weig...

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Published in:	The Journal of geometric analysis Vol. 33; no. 9
Main Author:	Shayya, Bassam
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.09.2023
Subjects:	Abstract Harmonic Analysis Convex and Discrete Geometry Differential Geometry Dynamical Systems and Ergodic Theory Fourier Analysis Global Analysis and Analysis on Manifolds Mathematics Mathematics and Statistics 28A75 Fourier restriction Polynomial partitioning Extension operator 42B20 42B10 Weighted restriction estimates
ISSN:	1050-6926, 1559-002X
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Abstract	Suppose 0 < α ≤ n , H : R n → [ 0 , 1 ] is a Lebesgue measurable function, and A α ( H ) is the infimum of all numbers C for which the inequality ∫ B H ( x ) d x ≤ C R α holds for all balls B ⊂ R n of radius R ≥ 1 . After Guth introduced polynomial partitioning to Fourier restriction theory, weighted restriction estimates of the form ‖ E f ‖ L p ( B , H d x ) ≲ R ϵ A α ( H ) 1 / p ‖ f ‖ L q ( σ ) have been studied and proved in several papers, leading to new results about the decay properties of spherical means of Fourier transforms of measures and, in some cases, to progress on Falconer’s distance set conjecture in geometric measure theory. This paper improves on the known estimates when E is the extension operator associated with the unit paraboloid P ⊂ R 3 , reaching the full possible range of p , q exponents (up to the sharp line) for p ≥ 3 + ( α - 2 ) / ( α + 1 ) and 2 < α ≤ 3 .
AbstractList	Suppose 0 < α ≤ n , H : R n → [ 0 , 1 ] is a Lebesgue measurable function, and A α ( H ) is the infimum of all numbers C for which the inequality ∫ B H ( x ) d x ≤ C R α holds for all balls B ⊂ R n of radius R ≥ 1 . After Guth introduced polynomial partitioning to Fourier restriction theory, weighted restriction estimates of the form ‖ E f ‖ L p ( B , H d x ) ≲ R ϵ A α ( H ) 1 / p ‖ f ‖ L q ( σ ) have been studied and proved in several papers, leading to new results about the decay properties of spherical means of Fourier transforms of measures and, in some cases, to progress on Falconer’s distance set conjecture in geometric measure theory. This paper improves on the known estimates when E is the extension operator associated with the unit paraboloid P ⊂ R 3 , reaching the full possible range of p , q exponents (up to the sharp line) for p ≥ 3 + ( α - 2 ) / ( α + 1 ) and 2 < α ≤ 3 .
Author	Shayya, Bassam
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Cites_doi	10.1007/s00039-011-0140-9 10.4007/annals.2015.182.1.9 10.1353/ajm.2021.0005 10.1215/S0012-7094-99-09610-2 10.1090/jams827 10.1112/plms.12046 10.1017/S0013091521000201 10.1007/s00039-018-0466-7 10.1090/S0894-0347-98-00278-1 10.1007/s11511-006-0006-4 10.4007/annals.2019.189.3.4 10.4171/RMI/188 10.4310/ACTA.2018.v221.n1.a3 10.1007/BF01896376
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Copyright	Mathematica Josephina, Inc. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
Copyright_xml	– notice: Mathematica Josephina, Inc. 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
DOI	10.1007/s12220-023-01364-0
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References	Bourgain (CR2) 1991; 1 Bourgain, Demeter (CR3) 2015; 182 Bennett, Carbery, Tao (CR1) 2006; 196 Guth (CR9) 2016; 29 CR6 CR5 Du, Zhang (CR8) 2019; 189 Shayya (CR13) 2017; 115 Guth (CR10) 2018; 221 CR17 Du, Guth, Ou, Wang, Wilson, Zhang (CR7) 2021; 143 Katz, Rogers (CR11) 2018; 28 Wolff (CR18) 1995; 11 CR12 Tao (CR15) 1999; 96 Shayya (CR14) 2021; 64 Bourgain, Guth (CR4) 2011; 21 Tao, Vargas, Vega (CR16) 1998; 11
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SubjectTerms	Abstract Harmonic Analysis Convex and Discrete Geometry Differential Geometry Dynamical Systems and Ergodic Theory Fourier Analysis Global Analysis and Analysis on Manifolds Mathematics Mathematics and Statistics
Title	Improved Weighted Restriction Estimates in R3
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