Projective limits of Poletsky--Stessin Hardy spaces

Poletsky-Stessin Hardy Spaces on Complex Ellipsoids in C^n

We study Poletsky-Stessin Hardy spaces on complex ellipsoids in C^n. Different from one variable case, classical Hardy spaces are strictly contained in Poletsky-Stessin Hardy spaces on complex ellipsoids so boundary values are not automatically obtained in this case. We have showed that functions belonging to Poletsky-Stessin Hardy spaces have boundary values and they can be approached through admissible approach regions in the complex ellipsoid case. Moreover, we have obtained that polynomials are dense in these spaces. We also considered the composition operators acting on Poletsky-Stessin H…

Poletsky-Stessin Hardy Spaces on Domains Bounded by An Analytic Jordan Curve in $\mathbb{C}$

We study Poletsky-Stessin Hardy spaces that are generated by continuous, subharmonic exhaustion functions on a domain $Ω\subset\mathbb{C}$, that is bounded by an analytic Jordan curve. Different from Poletsky & Stessin's work these exhaustion functions are not necessarily harmonic outside of a compact set but have finite Monge-Ampére mass. We have showed that functions belonging to Poletsky-Stessin Hardy spaces have a factorization analogous to classical Hardy spaces and the algebra $A(Ω)$ is dense in these spaces as in the classical case ; however, contrary to the classical Hardy spaces, comp…

Cauchy-Fantappie Type Operators And Duality On Poletsky-Stessin Hardy Spaces of Complex Ellipsoids

In the first part of this study we consider the boundedness and compactness properties of Cauchy-Fantappie type operators on Poletsky-Stessin Hardy spaces $H^{p}_{u}(\mathbb{B}^{\textbf{p}})$ of complex ellipsoids. We show that boundedness and compactness criteria are given by the Carleson conditions. In addition we give a basic compactness property for the subsets of $H^{p}_{u}(\mathbb{B}^{\textbf{p}})$ spaces and the characterization of weakly convergent sequences in $H^{p}_{u}(\mathbb{B}^{\textbf{p}})$. In the second part we will discuss the dual complement of the complex ellipsoid and we w…

Beurling-Type Invariant Subspaces of the Poletsky-Stessin Hardy Spaces in the Bidisc

The invariant subspaces of the Hardy space on $H^2(\mathbb{D})$ of the unit disc are very well known however in several variables the structure of the invariant subspaces of the classical Hardy spaces is not yet fully understood. In this study we examine the invariant subspace problem for Poletsky-Stessin Hardy spaces which is a natural generalization of the classical Hardy spaces to hyperconvex domains in $\mathbb{C}^n$. We showed that not all invariant subspaces of $H^{2}_{\tilde{u}}(\mathbb{D}^2)$ are of Beurling-type. To characterize the Beurling-type invariant subspaces of this space we f…

Poletsky-Stessin-Hardy spaces in the plane

In this paper we give two complete characterizations of the Poletsky- Stessin- Hardy spaces in the complex plane: First in terms of their boundary values as a weighted subclass of the usual $L^p$ class with respect to the arclength measure on the boundary. Second we completely describe functions in these spaces by having a harmonic majorant with a certain growth condition and we prove some basic results about these spaces. In particular, we prove approximation results in such spaces and extend the classical result of Beurling which describes the invariant subspaces of the shift operator. Addit…

Poletsky-Stessin 加权 Bergman 空间中的循环性

Cyclicity in Poletsky-Stessin Weighted Bergman Spaces

本文研究单位球、双圆盘和复椭球上 Poletsky-Stessin 加权 Bergman 空间中多项式的循环性，通过扩展参数范围构造类 Dirichlet 型空间，揭示与已有研究的差异并提出若干公开问题。