## UNLV Theses, Dissertations, Professional Papers, and Capstones

12-1-2020

Dissertation

#### Degree Name

Doctor of Philosophy (PhD)

#### Department

Mathematical Sciences

Zhijian Wu

Angel Muleshkov

Xin Li

Stephen Lepp

86

#### Abstract

In this dissertation, we introduce a Nevanlinna-type counting function, $M_{\varphi, \alpha}$. We show that $M_{\varphi, \alpha}$ is subharmonic and continuous on $\theset$ by exhibiting certain properties of the logarithmic potential of the associated pullback measure of two-dimensional Lebesgue measure on $\mathbb{D}$. Different identities of $M_{\varphi, \alpha}$ are presented. A change-of-variable formula calculates precisely the weighted Bergman norms of composition functions in terms of $M_{\varphi, \alpha}$. Certain estimates involving $M_{\varphi, \alpha}$ and the associated pullback measure on Carleson windows and semi-disks are demonstrated. For weighted Bergman space $A^2_\alpha$, we characterize orthogonal self-maps and give the Hilbert Schmidt norm of $C_\varphi$.

#### Keywords

Distribution; Hardy space; Potential theory; Radial; Super harmonic; Weighted Bergman space

Mathematics

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