  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.
