To the theory of generalized Stieltjes transforms

TL;DR

This paper advances the understanding of generalized Stieltjes transforms by characterizing measures, providing optimal estimates, and establishing criteria for positivity, thereby enhancing the theory and applications in spectral analysis and probabilistic contexts.

Contribution

It identifies measures in generalized Stieltjes transforms, offers optimal size estimates, and introduces criteria for measure positivity, addressing open problems and improving existing results.

Findings

Measures in product representations are themselves generalized Stieltjes transforms.

Optimal bounds for the size of these measures are established.

New conditions for representing functions as generalized Stieltjes transforms are derived.

Abstract

We identify measures arising in the representations of products of generalized Stieltjes transforms as generalized Stieltjes transforms, provide optimal estimates for the size of those measures, and address a similar issue for generalized Cauchy transforms. In the latter case, in two particular settings, we give criteria ensuring that the measures are positive. On this way, we also obtain new, applicable conditions for representability of functions as generalized Stieltjes transforms, thus providing a partial answer to a problem posed by Sokal and shedding a light at spectral multipliers emerged recently in probabilistic studies. As a byproduct of our approach, we improve several known results on Stieltjes and Hilbert transforms.