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