Weighted Hardy Spaces on the Unit Disk

Abstract

Poletsky and Stessin introduced (Indiana Univ Math J 57:2153–2201, 2008) weighted Hardy spaces $$H^p_u(D)$$ on a hyperconvex domain $$D$$ in $$\mathbb {C}^n$$ . For their definitions they used a plurisubharmonic exhaustion function $$u$$ on $$D$$ and related measures $$\mu _{u,r}$$ . In this paper we study such spaces when the domain $$D$$ is the unit disk $$\mathbb {D}$$ . We show that if the exhaustions are chosen so that the total mass of their Laplacian is 1, then the intersection of the unit balls $$B_{u,p}(1)$$ in $$H^p_u(\mathbb {D})$$ as $$u$$ ranges over all such exhaustions is the unit ball $$B_\infty (1)$$ in $$H^\infty (\mathbb {D})$$ . Demailly (Math Z 194:519–564, 1987) has proved that the measures $$\mu _{u,r}$$ converge weak- $$*$$ in $$C^*(\overline{D})$$ to a non-negative boundary measure $$\mu _u$$ as $$r\rightarrow 0^-$$ . We show that these measures converge weak- $$*$$ to $$\mu _u$$ also in the space dual to the weighted space $$h^p_u(\mathbb {D})$$ of harmonic functions. For the function $$f \in H^p_u(\mathbb {D})$$ , we define the dilations $$f_t(z) = f(tz), 0 < t < 1,$$ and prove that these dilations converge to the function $$f$$ in the $$H^p_u$$ -norm.