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Weak-Type Boundedness of the Fourier Transform on Rearrangement Invariant Function Spaces

Part of: Linear function spaces and their duals Harmonic analysis in several variables

Published online by Cambridge University Press:  08 June 2018

Santiago Boza  and
Javier Soria
Santiago Boza
Affiliation:
Department of Mathematics, EETAC, Polytechnical University of Catalonia, E-08860 Castelldefels, Spain (santiago.boza@upc.edu)
Javier Soria
Affiliation:
Department of Mathematics and Informatics, University of Barcelona, Gran Via 585, E-08007 Barcelona, Spain (soria@ub.edu)
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Abstract

We study several questions about the weak-type boundedness of the Fourier transform ℱ on rearrangement invariant spaces. In particular, we characterize the action of ℱ as a bounded operator from the minimal Lorentz space Λ(X) into the Marcinkiewicz maximal space M(X), both associated with a rearrangement invariant space X. Finally, we also prove some results establishing that the weak-type boundedness of ℱ, in certain weighted Lorentz spaces, is equivalent to the corresponding strong-type estimates.

Keywords

Fourier transformrearrangement invariant function spacesLorentz spacesweights

MSC classification

Primary: 46E30: Spaces of measurable functions ($L^p$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Secondary: 42B35: Function spaces arising in harmonic analysis
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 61 , Issue 3 , August 2018 , pp. 879 - 890
Copyright
Copyright © Edinburgh Mathematical Society 2018 

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