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Cauchy-Fantappie Type Operators And Duality On Poletsky-Stessin Hardy Spaces of Complex Ellipsoids
| Content Provider | Semantic Scholar |
|---|---|
| Author | Şahin, Sibel Yilmaz |
| Copyright Year | 2014 |
| 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 Hp u(B ) 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 Hp u(B ) spaces and the characterization of weakly convergent sequences in Hp u(B ). In the second part we will discuss the dual complement of the complex ellipsoid and we will give a duality result for Hp u(B ) spaces in the sense of Grothendieck-Köthe-da Silva. Introduction The aim of this paper is to study the behavior of Cauchy-Fantappie type operators on Poletsky-Stessin Hardy spaces [6] and to give a duality result for these spaces in the sense of [2]. In [8], [9] the Cauchy-Fantappie projection associated with Monge-Ampère measure is considered in relation with boundary values of PoletskyStessin Hardy spaces and Carleson measures. In this study we will examine the general Cauchy-Fantappie type operators in relation with Carleson measures. An analogous study has been done in [1] for Toeplitz operators in the setting of classical Bergman spaces of strictly pseudoconvex domains. In this paper we will work with the much more general holomorphic function spaces, namely Poletsky-Stessin Hardy spaces and structurally much more complex domains, the complex ellipsoids. The organization of this paper is as follows: In Section 1, we recall the PoletskyStessin Hardy spaces, H u(B ), on the complex ellipsoid B and we introduce the Cauchy-Fantappie integral associated with the Monge-Ampère measure μu, together with an integral representation for H u(B ). The main results of this study are given in the following sections: In Section 2, we first introduce general Cauchy-Fantappie type operator associated with a finite Borel measure μ and we give boundedness and compactness criteria for these operators in terms of Carleson measures. In addition we will give a discussion of a compactness property for subsets of H u(B ) and a characterization of vanishing Carleson measures of H u(B ). Finally, in Section 3, we first give a brief introduction about Grothendieck-Köthe-da Silva duality for the spaces of holomorphic functions defined in a convex domain and then using a general characterization of dual complements of Reinhardt domains [2] we give the dual complement of some special type of complex ellipsoids. Finally, we prove a duality result for Poletsky-Stessin Hardy space of complex ellipsoid. Date: February 6, 2015. |
| File Format | PDF HTM / HTML |
| Alternate Webpage(s) | https://arxiv.org/pdf/1412.2348v3.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
