Positive definite matrices and integral equations on unbounded domains

Abstract

The kernel of a continuous positive integral operator on an interval $I$ is a Moore matrix on $I$. We show that, under minimal differentiability assumptions, this implies that the kernel satisfies a 2-parameter family of differential inequalities. These inequalities ensure that, for unbounded $I$, the corresponding integral operator is exceptionally well behaved: it is compact and thus the eigenfunctions for its discrete spectrum have the differentiability of the kernel and satisfy sharp Sobolev bounds, the symmetric mixed partial derivatives are again kernels of positive operators and the differentiated eigenfunction series converge uniformly and absolutely. Converse results are derived.