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.