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Weighted Hardy spaces on the unit disk. (English) Zbl 1325.30051

Summary: E. A. Poletsky and M. I. Stessin [Indiana Univ. Math. J. 57, No. 5, 2153–2201 (2008; Zbl 1160.32008)] introduced 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 \(H^\infty(\mathbb D)\). J.-P. Demailly [Math. Z. 194, 519–564 (1987; Zbl 0595.32006)] has proved that the measures \(\mu _{u,r}\) converge weak-\(*\) in \(C^*(\overline{D})\) to a non-negative boundary measure \(\mu _u\) as \(r\to 0^-\). 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.

MSC:

30H10 Hardy spaces
30J99 Function theory on the disc
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
30E25 Boundary value problems in the complex plane
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