Weighted estimates for integral transforms and a variant of Schur's Lemma

In this paper we consider modular inequalities for integral transforms. We provide a variant of Schur's Lemma and establish sufficient conditions for modular inequalities to hold. Estimates for the constants are also given. As applications, the boundedness and estimates for several averaging operators and related integral transforms are obtained, which include the Laplace transform, the modified Lambert transform, the Stieltjes transform, the Hardy-type integral transforms and related geometric mean operators. We show that several known results in the literatures can be obtained by our results.