On analytic multifunctions

For any multifunction S ? D _z × ? _w , we give a criterion for analyticity (pseudoconcavity) in terms of plurisubharmonicity of the function V(z, w) = ?lnρ(w, S _z ), where ρ(w, S _a ) stands for the distance from the point w to the set S _a = S ∩ {z = a}.     

Continuation of separately analytic functions defined on part of the domain boundary

Let D ? ? ⁿ be a domain with smooth boundary ?D, let E??D be a subset of positive Lebesgue measure mes(E) > 0, and let F ? G be a nonpluripolar compact set in a strongly pseudoconvex domain D ? ? ^m . We prove that, under an additional condition, each function separately analytic on the set X = (D × F) ∪ (E × G) has a holomorphic contination to the domain $\rlap{--} X = \{ (z,w) \in D \times G:\omega _{in}^ * (z,E,D) + \omega ^ * (w,F,D) < 1\} $ , where ω* is the P-measure and ω*_in is the interior P-measure.     

Continuation of separately analytic functions defined on part of a domain boundary

Suppose that D ? ?ⁿ is a domain with smooth boundary ?D, E ? ?D is a boundary subset of positive Lebesgue measure mes(E) > 0, and F ? G is a nonpluripolar compact set in a strongly pseudoconvex domain G ? ?^m. We prove that, under some additional conditions, each function separately analytic on the set X = (D×F)∪(E× G) can be holomorphically continued into the domain where ω* is the P-measure and ω _in ^* is the inner P-measure.     

Subsets of full measure in a generic submanifold in \mathbb C ^n are non-plurithin

In this paper we prove that if $I\subset M $ is a subset of measure $0$ in a $C^2$ -smooth generic submanifold $M \subset \mathbb C ^n$ , then $M \setminus I$ is non-plurithin at each point of $M$ in $\mathbb C ^n$ . This result improves a previous result of A. Edigarian and J. Wiegerinck who considered the case where $I$ is pluripolar set contained in a $C^1$ -smooth generic submanifold $M \subset \mathbb C ^n$ (Edigarian and Wiegernick in Math. Z. 266(2):393–398, 2010). The proof of our result is essentially different.     

Potential theory in the class of m-subharmonic functions

A potential theory for the equation (dd ^c u)m ∧ β ^n?m = fβ ⁿ , 1 ≤ m ≤ n, is developed. The corresponding notions of m-capacity and m-subharmonic functions are introduced, and their properties are studied.     

Extension of holomorphic and pluriharmonic functions with thin singularities on parallel sections

The paper is of survey character. We present and discuss recent results concerning the extension of functions that admit holomorphic or plurisubharmonic extension in a fixed direction. These results are closely related to Hartogs’ fundamental theorem, which states that if a function f(z), z = (z ₁, z ₂, ..., z _z ), is holomorphic in a domain D ? ?ⁿ in each variable z _j , then it is holomorphic in D in the n-variable sense.