Weighted Hardy Spaces on the Unit Disk

Structure of weighted Hardy spaces on finitely connected domains

“…(3) We describe an isomorphism from the classical Hardy space H p (G) onto the space H p u (G). The last two results extend the research in [2], [11] and [12] from the disk to finitely connected planar domains. This paper is organized as follows: Section 2 is a brief summary of the previous work which are related and will be used in the paper.…”

“…By definition to each subharmonic function u continuous near the boundary of G corresponds a space, which is denoted by H p u of holomorphic functions in G. Here G is a bounded regular domain in C. Throughout the paper these spaces will be called Poletsky-Stessin Hardy spaces. Following the motivating work of Poletsky and Stessin, the structure and first examples on the unit disk of Poletsky-Stessin Hardy spaces were further investigated in [2], [11] and [12]. Among these recent work, the author and M. A. Alan gave a complete characterization of H p u spaces that live in the plane domains through the boundary values of the functions in this class or through a growth description of their harmonic majorants.…”

A New Class of Hardy Spaces in the Plane

“…is the balayage of the positive measure ∆u to the boundary ∂G. Then Vu(ζ) = ∂u ∂n (ζ) is the directional derivative of u in the normal direction at a point ζ ∈ ∂G (see [4] and [11]). The next results are restatements from [9] and they establish basic observations on the classes of Hardy spaces.…”

“…In the denition of W p we make use of the recently studied Poletsky-Stessin-Hardy (PSH) spaces. These spaces were introduced in several complex variables context in [9] and recently studied in planar domains in [1] and for the disk in the papers [10] and [11].…”