Boundary Values Properties of Functions in Weighted Hardy Spaces

Abstract

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.

Weighted Hardy spaces on the unit disk

“…M. A. Alan and N. G. Goguş in [1], S. S ¸ahin in [10] and K. R. Shrestha in [9] have independently produced examples that in general these new spaces H p u (D) do not coincide with the classical spaces H p (D). They have found an exhaustion funcion u for which H p u (D) H p (D).…”

“…Recently these spaces were subjected to more detailed studies in [1,9,10] when D is the unit disk D. The main goal of these papers was to establish properties of weighted Hardy spaces similar to the standard properties of the classical Hardy spaces. In this paper we continue this program looking more thoroughly at the properties of boundary values.…”

“…Hence every function f in H p u (D) or in h p u (D) has radial boundary values f * almost everywhere on T = ∂D with respect to the Lebesgue measure. It was proved in [2] that the measures µ u,r converge weak- * in C * (D) to a measure µ u supported by T when r → 0 − and it was established in [1,9,10] that…”

On weighted Hardy spaces on the unit disk

“…Recently, M. Alan and N. Gogus in [1], S. Sahin in [9] and K. R. Shrestha in [10,11] obtained the description of these spaces on the unit disk and called them Poletsky-Stessin Hardy spaces. In particular, they showed that the new spaces form a subclass of weighted Hardy spaces introduced at the beginning of this section.…”

Projective Limits of Poletsky–Stessin Hardy Spaces

“…Recently, M. Alan and N. Gogus in [1], S. Sahin in [14], K. R. Shrestha in [16] and the latter with the author in [11] showed that if D is the unit disk D these spaces form a subclass of weighted Hardy spaces studied, for example, in [8] and [3]. However, these subclass has special properties and, moreover, has no analogs in several variables.…”