On the theory of generalized Stieltjes transforms

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. In 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 on spectral multipliers emerged recently in probabilistic studies. As a byproduct of our approach, we improve several known results on Stieltjes and Hilbert transforms.