Weighted Fourier Inequalities in Lebesgue and Lorentz Spaces

In this paper, we obtain sufficient conditions for the weighted Fourier-type transforms to be bounded in Lebesgue and Lorentz spaces. Two types of results are discussed. First, we review the method based on rearrangement inequalities and the corresponding Hardy’s inequalities. Second, we present Hör...

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Bibliographic Details
Published in:The Journal of fourier analysis and applications 2020-08, Vol.26 (4), Article 57
Main Authors: Nursultanov, Erlan, Tikhonov, Sergey
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Approximations and Expansions
Fourier Analysis
Inequalities
Mathematical Methods in Physics
Mathematics
Mathematics and Statistics
Operators (mathematics)
Partial Differential Equations
Signal,Image and Speech Processing
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Summary:In this paper, we obtain sufficient conditions for the weighted Fourier-type transforms to be bounded in Lebesgue and Lorentz spaces. Two types of results are discussed. First, we review the method based on rearrangement inequalities and the corresponding Hardy’s inequalities. Second, we present Hörmander-type conditions on weights so that Fourier-type integral operators are bounded in Lebesgue and Lorentz spaces. Both restricted weak- and strong-type results are obtained. In the case of regular weights necessary and sufficient conditions are given.
ISSN:1069-5869
1531-5851
DOI:10.1007/s00041-020-09764-4