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The Fourier transform in weighted spaces. (English) Zbl 0688.42010

Approximation and function spaces, Proc. 27th Semest., Warsaw/Pol. 1986, Banach Cent. Publ. 22, 173-182 (1989).
[For the entire collection see Zbl 0681.00013.]
This paper recounts some recent results about mapping properties of the Fourier transform in weighted spaces. For example, conditions on the weight functions u and v are imposed for which the Fourier transform is bounded from \(L^ p_ v\) to \(L^ q_ u\), \(0<p,q<\infty\), \(p>1\). Under additional monotonicity conditions the results are sharp in the range \(1<p\leq q<\infty\). In the case \(0<p\leq 1\) mapping properties still hold provided the domain spaces are replaced by the weighted Hardy spaces. Some applications are given to illustrate the results.
Reviewer: H.P.Heinig

MSC:

42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
42-06 Proceedings, conferences, collections, etc. pertaining to harmonic analysis on Euclidean spaces

Citations:

Zbl 0681.00013