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The Fourier transform in weighted Lorentz spaces. (English) Zbl 1045.42004

Summary: Necessary conditions and sufficient conditions on weights \(u\) and \(w\) are given for the Fourier transform \({\mathcal F}\) to be bounded as a map between the Lorentz spaces \(\Gamma_q(w)\) and \(\Lambda_p (u)\). This may be viewed as a weighted extension of a result of Jodeit and Torchinsky on operators of type \((1,\infty)\) and (2,2). In the case \(0<p \leq 2=q\), the necessary and sufficient conditions are equivalent and give a simple weight condition which is equivalent to \({\mathcal F}:\Gamma_2(w) \to \Lambda_p(u)\) and also to \({\mathcal F}:\Gamma_2(w)\to \Gamma_p (u)\).

MSC:

42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42B35 Function spaces arising in harmonic analysis
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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