Vector-valued functions integrable with respect to bilinear maps
Let \((\Omega,\Sigma,\mu)\) be a \(\sigma\)-finite complete measure space, \(1\leq p<\infty\), and let \(X,Y,Z\) be Banach spaces. Suppose \({\mathcal B}:X\times Y\to Z\) is a bounded bilinear map. It is introduced the notion of an \(X\)-valued function \(p\)-integrable with respect to \({\mathcal B}\): a function \(f:\Omega\to X\) is such whenever \(\sup\left\{\int_\Omega\|({\mathcal B}f(w),y)\|^p d\mu(w):\|y\|=1\right\}<\infty\). The spaces of functions integrable with respect to bilinear maps arising from the Hölder and the Young inequalities are identified. These results are applied to establish conditions on \(X\)-valued kernels \(k\) for the boundedness of integral operators \(T_k^{\mathcal B}: L^p(\Omega',Y)\to L^p(\Omega,Z)\) given by \(T_k^{\mathcal B}(g)(w)=\int_{\Omega'}{\mathcal B}(k(w,w'),g(w'))d\mu'(w')\), where \((\Omega',\Sigma',d\mu')\) is also a \(\sigma\)-finite complete measure space.
scientific article; zbMATH DE number 5567837
scientific article