Carleson measures in Hardy and weighted Bergman spaces of polydiscs

Abstract

The importance of theorems on Carleson measures has been well recognized [3]. In [1] Chang has given a characterization of the bounded measures on L p ( T n ) {L^p}({T^n}) using what one may characterize as the bounded identity operators from Hardy spaces of polydiscs in L p {L^p} spaces. In [4] Hastings gives a similar result for (unweighted) Bergman spaces of polydiscs. In this paper we characterize the bounded identity operators from weighted Bergman spaces of polydiscs into L p {L^p} spaces, and classify those operators which are compact on the Hardy and weighted Bergman spaces in terms of Carleson-type conditions. We give two immediate applications of these results here, and a much broader class of applications elsewhere [5].