Transferring estimates for zonal convolution operators (Q909187)

Let G be a unimodular locally compact group with a compact subgroup K and a unimodular subgroup H such that \(G=KHK\), and the Haar integral of f on G can be written as \(\int_{K}\int_{H}\int_{K}f(k'xk)w(x)dk'dxdk\). Then a K-invariant function (or distribution) \(\phi\) on G is a convolver of \(L^ p(G)\) if \(w\cdot \phi |_ H\) is a convolver of \(L^ p(H)\), and certain norm estimates hold. This was proved (in less generality) by \textit{R. Coifman} and \textit{G. Weiss} using ``transference'' [Transference methods in analysis (C.B.M.S. Conf. Ser. 31, Am. Mat. Soc., 1977; Zbl 0377.43001]. In the paper under review, the general result is proved using the spaces \(A_ p(G)\), introduced by \textit{A. Figà-Talamanca} [Duke Math. J. 32, 495-501 (1965; Zbl 0142.104)] and shown to be algebras by \textit{C. S. Herz} [C. R. Acad. Sci., Paris 260, 6001-6004 (1965; Zbl 0135.354)]. Various applications are given.