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On Dirichlet Spaces With a Class of Superharmonic Weights

Published online by Cambridge University Press:  20 November 2018

Guanlong Bao ,
Nihat Gokhan Göğüş  and
Stamatis Pouliasis
Guanlong Bao
Affiliation:
Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China email: [email protected]
Nihat Gokhan Göğüş
Affiliation:
Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, Istanbul, 34956 Turkey email: [email protected]@gmail.com
Stamatis Pouliasis
Affiliation:
Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, Istanbul, 34956 Turkey email: [email protected]@gmail.com
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Abstract

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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 ${{D}_{\mu }}$.

Keywords

30H1031C2546E15Dirichlet spaceHardy spacesuperharmonic weight
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 70 , Issue 4 , 01 August 2018 , pp. 721 - 741
Copyright
Copyright © Canadian Mathematical Society 2018

References

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