An integral operator inequality with applications

Linear integral operators are defined acting in the Lebesgue integration sp aces on intervals of the real line. A necessary and sufficient condition is given for these operators to be bounded, and a characterisation is given f or the operator bounds. There are applications of the results to integral i nequalities; also to properties of the domains of self-adjoint unbounded op erators, in Hilbert function spaces, associated with the classical orthogon al polynomials and their generalisations.