Poletsky–Stessin–Hardy Spaces in the Plane

A New Class of Hardy Spaces in the Plane

“…Let u ∈ E(G) and p ≥ 1 (p > 0 if G is simply connected). Then the space H p u (G) (thinking of boundary values) is a closed subspace of the weighted space L p (Vudσ) on the boundary ∂G, where (see [1]) dσ is the usual Lebesgue measure on ∂G and…”

“…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].…”

On weighted Hardy spaces on the unit disk

“…By Corollary 3.2 we can replace the lim in the above definition with lim. By Theorem 3.1 and the monotone convergence theorem it follows that, It has been established (see [1], [9] and [10]) that the boundary measure…”

Structure of weighted Hardy spaces on finitely connected domains

“…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.…”

“…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. As an application, a Beurling's type theorem was proved in [2]. This states roughly that if G is the unit disk D, and u is such a subharmonic exhaustion on D, there exists a bounded outer function φ so that the space H p u isometrically equals to M φ,p , p > 0.…”