On Dirichlet Spaces With a Class of Superharmonic Weights. (1st August 2018)

• Record Type: Journal Article
• Authors: Bao, Guanlong; Göğüş, Nihat Gokhan; Pouliasis, Stamatis
• Is Part Of: Canadian journal of mathematics. Volume 70:Number 4(2018)
• Page Start: 721
• Page End: 741
• Publication Date: 2018-08-01
• Subjects: 30H10 -- 31C25 -- 46E15; Dirichlet space -- Hardy space -- superharmonic weight; Mathematics -- Periodicals; Mathematics; Electronic journals; Periodicals; 510
• DOI: 10.4153/CJM-2017-005-1
• Languages: English
• ISSNs: 0008-414X

Abstract

Abstract: In this paper, we investigate Dirichlet spaces ${{D}_{\mu }}$ with superharmonic weights induced by positive Borel measures $\mu $ on the open unit disk. We establish the Alexander-Taylor-Ullman inequality for ${{D}_{\mu }}$ spaces and we characterize the cases where equality occurs. We define a class of weighted Hardy spaces $H_{\mu }^{2}$ via the balayage of the measure $\mu $ . We show that ${{D}_{\mu }}$ is equal to $H_{\mu }^{2}$ if and only if $\mu $ is a Carleson measure for ${{D}_{\mu }}$ . As an application, we obtain the reproducing kernel of ${{D}_{\mu }}$ when $\mu $ is an infinite sum of point-mass measures. We consider the boundary behavior and innerouter factorization of functions in ${{D}_{\mu }}$ . We also characterize the boundedness and compactness of composition operators on ${{D}_{\mu }}$ .