In this paper we study the boundary values of harmonic and holomorphic functions in the weighted Hardy spaces on the unit disk D. These spaces were introduced by Poletsky and Stessin in [6] for plurisubharmonic functions on hyperconvex domains D ⊂ C n as generalizations of classical Hardy spaces. We show that in the case when D is the unit disk D the theory of boundary values for functions in these spaces is analogous to the classical one.

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In this paper we completely characterize those weighted Hardy spaces that are Poletsky-Stessin Hardy spaces H p u . We also provide a reduction of H ∞ problems to H p u problems and demonstrate how such a reduction can be used to make shortcuts in the proofs of the interpolation theorem and corona problem.

In this paper we mainly discuss three things. First, there is no canonical norm on the space H p u (D). Second, we improve the weak- * convergence of the measures µu,r. Third, the dilations ft of the function f ∈ H p u (D) converge to f in H p u -norm and hence the polynomials are dense in H p u (D).