On the Schur test for \(L_2\)-boundedness of positive integral operators with a Wiener-Hopf example
Let \((\Omega,{\mathcal M},\mu)\), \((\Gamma,{\mathcal N},\nu)\) be \(\sigma\)-finite measure spaces, let \(K:\Gamma\times \Omega\to(0,\infty)\) be a measurable function on the product space, and let \({\mathcal D}(K)\) be the linear space of measurable functions for which the operator \({\mathcal X}\) is defined by \[ {\mathcal X}(f)(x)= \int_\Omega K(x,y) f(y)d\mu(y). \] In addition, let \({\mathcal D}(K')\) be the linear space of measurable functions for which the adjoint operator \({\mathcal X}'\) is defined by \[ {\mathcal X}'(g)(y)= \int_\Gamma K(x,y) g(x)d\nu(x). \] The norm in \(L^p(\Omega)\) (or \(L^p(\Gamma)\)) consisting of \(p\)th power integrable functions is denoted by \(\|\cdot\|_p\), the norm of an operator \(\mathbb{P}\) which is bounded from \(L^p\) into an \(L^q\)-space is denoted by \(\| \mathbb{P}\|_{p\to q}\), and the norm of an operator \({\mathcal T}\) which is bounded from a Banach space \(X\) into a Banach space \(Y\) is denoted by \(\|{\mathcal T}\|_{X\to Y}\). The main results of this paper are stated as propositions, which include: (1) \({\mathcal X}: L^2(\Omega)\to L^2(\Gamma)\) is a bounded linear operator if and only if there is a linear space \(X\) consisting of measurable functions on \(\Omega\), such that \({\mathcal X}\circ{\mathcal X}\) is a bounded linear operator in \(X\) with \(\|{\mathcal X}\|^2_{2\to 2}\leq\|{\mathcal X}'\circ{\mathcal X}\|_{X\to X}\); and \(\|{\mathcal X}\|^2_{2\to 2}=\|{\mathcal X}'\|^2_{2\to 2}=\|{\mathcal X}'\circ{\mathcal X}\|^2_{2\to 2}\); (2) (Schur's condition) If \(h\in{\mathcal D}(K'\circ K)\) and finite constant \(A\) are such that \({\mathcal X}'\circ{\mathcal X}(h)(x)\leq A^2h(x)\) for almost all \(x\) in \(\Omega\), then \({\mathcal X}\) is a bounded operator with \(\|{\mathcal X}\|_{2\to 2}\leq A\); (3) if \({\mathcal X}\) is a bounded operator from \(L^2(\Omega)\) into \(L^2(\Gamma)\), then \[ \|{\mathcal X}\|_{2\to 2}= \inf\{\Lambda:{\mathcal X}'\circ{\mathcal X}(g)\leq\Lambda^2 g, g\in L^2(\Omega), g>0\text{ a.e.}\}. \] An example to which the general results are applicable involves a kernel \({\mathcal X}_\alpha\) associated with Wiener-Hopf theory.
scientific article; zbMATH DE number 1250658