Following Sibony, we say that a bounded domain in is -regular if every continuous real valued function on the boundary of can be extended continuously to a plurisubharmonic function on . The aim of this paper is to study an analogue of this concept in the category of unbounded domains in . The use of Jensen measures relative to classes of plurisubharmonic functions plays a key role in our work
In this note by using techniques similar to that of [2] and [3], we study the local polynomial convexity of perturbation of union of two totally real planes meeting along a real line.
We generalize classical results for plurisubharmonic functions and hyperconvex domain to q-plurisubharmonic functions and q-hyperconvex domains. We show, among other things, that B-regular domains are q-hyperconvex. Moreover, some smoothing results for q-plurisubharmonic functions are also given.
We give sufficient conditions for unicity of plurisubharmonic functions in Cegrell classes.
A subset K of ℂⁿ is said to be regular in the sense of pluripotential theory if the pluricomplex Green function (or Siciak extremal function) is continuous in ℂⁿ. We show that K is regular if the intersections of K with sufficiently many complex lines are regular (as subsets of ℂ). A complete characterization of regularity for Reinhardt sets is also given.
We study the weighted Bernstein-Markov property for subsets in ℂⁿ which might not be bounded. An application concerning approximation of the weighted Green function using Bergman kernels is also given.
Let D be a domain in ℂⁿ. We introduce a class of pluripolar sets in D which is essentially contained in the class of complete pluripolar sets. An application of this new class to the problem of approximation of holomorphic functions is also given.
Let F be the Cartesian product of N closed sets in ℂ. We prove that there exists a function g which is continuous on F and holomorphic on the interior of F such that is complete pluripolar in . Using this result, we show that if D is an analytic polyhedron then there exists a bounded holomorphic function g such that is complete pluripolar in . These results are high-dimensional analogs of the previous ones due to Edlund [Complete pluripolar curves and graphs, Ann. Polon. Math. 84 (2004), 75-86]...
The aim of the paper is to establish some results on pluripolar hulls and to define pluripolar hulls of certain graphs.
We study the B-regularity of some classes of domains in ℂⁿ. The results include a complete characterization of B-regularity in the class of Reinhardt domains, we also give some sufficient conditions for Hartogs domains to be B-regular. The last result yields sufficient conditions for preservation of B-regularity under holomorphic mappings.
We first establish the equivalence between hyperconvexity of a fat bounded Reinhardt domain and the existence of a Stein neighbourhood basis of its closure. Next, we give a necessary and sufficient condition on a bounded Reinhardt domain D so that every holomorphic mapping from the punctured disk into D can be extended holomorphically to a map from Δ into D.
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