Szeg\H o theorem, Carath\'eodory domains, and boundedness of calculating functionals

F. G. Abdullaev ; A. A. Dovgoshey

Matematičeskie zametki, Tome 77 (2005) no. 1, pp. 3-15

Résumé

Suppose that $G$ is a bounded simply connected domain on the plane with boundary $\Gamma$, $z_0\in G$, $\omega$ is the harmonic measure with respect to $z_0$, on $\Gamma$, $\mu$ is a finite Borel measure with support $\operatorname{supp}(\mu)\subseteq\Gamma$, $\mu_a+\mu_s$ is the decomposition of $\mu$ with respect to $\omega$, and $t$ is a positive real number. We solve the following problem: for what geometry of the domain $G$ is the condition $$ \int\ln\biggl(\frac{d\mu_a}{d\omega}\biggr)\,d\omega=-\infty $$ equivalent to the completeness of the polynomials in$L^t(\mu)$ or to the unboundedness of the calculating functional $p\to p(z_0)$, where $p$ is a polynomial in $L^t(\mu)$? We study the relationship between the densities of the algebras of rational functions in $L^t(\mu)$ and $C(\Gamma)$. For $t=2$, we obtain a sufficient criterion for the unboundedness of the calculating functional in the case of finite Borel measures with support of an arbitrary geometry.