Bilinear maps and convolutions (Q2762063)

Let \(G\) be a locally compact abelian group and let \(d\mu_G\) denote the Haar measure on \(G\). Let \(u:X\times Y\to Z\) be a bounded continuous bilinear map between Banach spaces. The author defines the \(u\)-convolution of \(f\in L^1(G,X)\) and \(g\in L^1(G,Y)\) by \(f*_ug(t): =\int_Gu(f(t-s)\), \(g(s))d \mu_G(s)\). A bilinear Marcinkiewicz-Zygmund theorem is proved. The \((u,X)\)-bounded approximation identities and the \((u,X)\)-summability kernels are defined for \(u:X\times Y\to X\), and the corresponding approximation result is proved. In the last section, a theorem of Hausdorff-Young type under certain assumptions on the Banach space is presented.NEWLINENEWLINEFor the entire collection see [Zbl 0970.00016].