부산대학교 도서관

For the classical Dirichlet space $$ \Cal{D} = \left\{f \in \roman{Hol}(\Bbb{D}): \int_{\bold{D}} |f'(z)|^2 dA < \infty\right\}, $$ it is known that a Blaschke product belongs to $\Cal{D}$ if and only if it is a finite Blaschke product. This paper explores the same question for weighted Dirichlet spaces $$ \Cal{D}_{\omega} = \left\{f \in \roman{Hol}(\Bbb{D}): \int_{\bold{D}} |f'(z)|^2 \omega(w) dA < \infty\right\}. $$ In the above, $\omega\: \Bbb{D} \to (0, \infty]$ is integrable with respect to area measure $dA$ on $\Bbb{D}$. One of the main theorems of this paper gives the following equivalence statement for a superharmonic weight function $\omega$: (i) $\Cal{D}_{\omega}$ contains no infinite Blaschke product; (ii) $\liminf_{|z| \to 1^{-}} \omega(z) > 0$; (iii) $\Cal{D}_{\omega} \subset \Cal{D}$. This paper also contains conditions on a sequence $(\lambda_n)_{n = 1}^{\infty} \subset \Bbb{D}$ to be the zeros of function from $\Cal{D}_{\omega} \setminus \{0\}$.