Moment Theory and Continuity of the Hilbert and Poisson Transforms in L2 Spaces

Abstract

Some moment problems, related to lifting properties and to questions concerning unconditional bases, lead to the study of a class (C) of measures μ in the unit disc, characterized by the property that the Poisson integral acts as a bounded operator from L2(dt) to L2(μ). On the other hand, problems concerning unconditional bases and from Prediction theory lead to a class (RN) of a measures μ in the unit circle, characterized by the property that the Hilbert transform acts continuously in the space {f ∈ L2(μ), f ^ ( n ) = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072577/94f683e7-9f6e-41e2-9c5e-98432cdb82df/content/eq253.tif"/> for n < N]. Both classes turned out to be closely related to the dual of the space H1. (A remarkable exposition of the topics concerning H1 and (c) is given in D. Sarason’s book [46]). In [15; 15a; 1; 2]), an attempt was made to establish a more explicit relation between moment theory and those classes. In fact there is a lifting property related to moment questions, which allows to deduce in a unified way basic facts concerning those classes and the dual of H1, with some refinements and new results. This moment approach still has to be fully developed, and it might be not totally out of place to review the above-mentioned questions. However, only the simplest notions will be recalled and no complete survey is intended.