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

TL;DR

This paper investigates the boundedness, compactness, and duality properties of Cauchy-Fantappie operators on Poletsky-Stessin Hardy spaces over complex ellipsoids, establishing Carleson condition criteria and duality results.

Contribution

It provides new criteria for boundedness and compactness of operators on these Hardy spaces and characterizes their dual spaces in the context of complex ellipsoids.

Findings

Boundedness and compactness characterized by Carleson conditions.

Established basic compactness properties and weak convergence criteria.

Derived duality results for Hardy spaces on complex ellipsoids.

Abstract

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 will give a duality result for $H^{p}_{u}(\mathbb{B}^{\textbf{p}})$ spaces in the sense of Grothendieck-K\"{o}the-da Silva.