Duality of Fourier type with respect to locally compact Abelian groups
The Hausdorff-Young inequality \(\|\widehat{f}\|_{q} \leq \|f\|_{p}, f \in L^{1}\cap L^{p}\), \(\widehat{f} = \mathcal{F}f\) the Fourier transform of \(f\), was initially related to ordinary Fourier series. Let \(X\) be a Banach space with dual denoted \(X'\) and \(G\) a locally compact Abelian group with dual denoted \(G'\). \textit{J. Peetre} [Rend. Sem. Mat. Univ. Padova 42, 15--26 (1969; Zbl 0241.46033)] extended Hausdorff-Young to Banach spaces of `Fourier type \(p\)', i.e., where the Fourier transform can be extended from the algebraic tensor product of \(L^{p}\) and \(X\) to become a bounded linear operator from \(L^{p}_{X}\) to \(L^{q}_{X}\). \textit{M. Milman} [Ann. Mat. Pura Appl., IV. Ser. 136, 317--328 (1984; Zbl 0578.46068)] extended the concept of `Fourier type' to functions on groups, the extension leading to an operator from \(L^{p}_{X}(G)\) to \(L^{q}_{X}(G')\). The Hausdorff-Young inequality can be expressed as \(\| \mathcal{F}_{G}f\| \leq c\|f\|\) for a real \(c= c(p,G)\), \(f \in L^{p}_{X}(G)\), the left hand side having the \(L^{q}_{X}(G')\) norm. This is shown to imply a dual inequality \(\|\mathcal{F}_{G'}{g}\| \leq c'\|g\|\), for all \(g \in L^{p}_{X'}(G')\), where the left hand side has the \(L^{q}_{X'}(G)\) norm; etc. This was known to be true for some groups, such as \(\mathbb{R}\), \(\mathbb{T}\), \(\mathbb{Z}\) and the Cantor group \(\{ 0,1 \} ^{n}\) [see \textit{J. Garcia-Cuerva, K. S. Kazarian, V. I. Kolyada} and \textit{J. L. Torrea}, Russ. Math. Surv. 53, 435--513 (1998; Zbl 0923.46017)]. The object of this paper is to prove all this is true for arbitrary locally compact Abelian \(G\). In their proof the authors first extend Milman's results to products of groups. Then they consider groups which have open subgroups; the proof reduces to that for discrete groups which turn out to be isomorphic to appropriate product groups.
scientific article; zbMATH DE number 5145476
scientific article