Abstract view
On Dirichlet spaces with a class of superharmonic weights


Guanlong Bao,
Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China
Nihat Gökhan Göğüş,
Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, Istanbul, 34956 Turkey
Stamatis Pouliasis,
Faculty of Engineering and Natural Sciences, Sabanci University, Tuzla, Istanbul, 34956 Turkey
Abstract
In this paper, we investigate Dirichlet spaces $\mathcal{D}_\mu$ with
superharmonic weights induced by positive Borel measures $\mu$
on
the open unit disk. We establish the AlexanderTaylorUllman
inequality for $\mathcal{D}_\mu$ 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 $\mathcal{D}_\mu$
is equal to $H_{\mu}^{2}$ if and only if $\mu$ is a
Carleson measure for $\mathcal{D}_\mu$. As an application, we obtain the
reproducing kernel of $\mathcal{D}_\mu$ when $\mu$ is an infinite
sum of point mass measures. We consider the boundary
behavior and innerouter factorization of functions in $\mathcal{D}_\mu$.
We also characterize the boundedness and
compactness of composition operators on $\mathcal{D}_\mu$.