Heinig, H. P. 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 Keywords:\(A_ p\)-weights; Heisenberg-Weyl inequality; Fourier transform in weighted spaces; weighted Hardy spaces; applications Citations:Zbl 0681.00013 PDFBibTeX XML