The Fourier transforms of Herz spaces on certain groups (Q792568)

Let G be a locally compact Abelian group with a suitable family of compact open subgroups in the sense of \textit{R. E. Edwards} and \textit{G. I. Gaudry} [''Littlewood-Paley and Multiplier Theory'' (1977; Zbl 0464.42013)] and let X be its dual group. Let \(K(\alpha\),p,q;G) and \(K(\alpha\),p,q;X) denote the Beurling-Herz spaces on G and X, as defined by the author in [Lect. Notes Math. 939, 106-121 (1981; Zbl 0507.46026)]. Theorem: For \(1<p\leq 2\) and \(0\leq \alpha<1-1/p\) the Fourier transform maps \(K(\alpha\),p,p;G) continuously into K(-\(\alpha\),p',2;X). The proof requires two results that are of independent interest: an extension of the Hausdorff-Young inequality to certain weighted \(L_ p\)- spaces on G and a generalization of the Littlewood-Paley theorem to the same weighted \(L_ p\)-spaces.