Evgeny A. Poletsky

Quasianalyticity and pluripolarity

be the graph of / in C2. The set Tf is always pluripolar when / is a real analytic function. In [DF], Diederich and Fornaess give an example of a C?? function / with nonpluripolar graph in C2. The paper [LMP] contains an example of a holomorphic function / on the unit disk U, continuous up to the boundary, such that the graph of / over S is not pluripolar as a subset of C2. Thus a priori the pluripolarity of graphs of functions on S is indeterminate. In this paper we prove that graphs of quasianalytic functions are still pluripolar (all necessary definitions can be found in the next section). More precisely,

Approximation by harmonic functions

For a compact set X ⊂ Rn we construct a restoring covering for the space h(X) of real-valued functions on X which can be uniformly approximated by harmonic functions. Functions from h(X) restricted to an element Y of this covering possess some analytic properties. In particular, every nonnegative function f ∈ h(Y ), equal to 0 on an open non-void set, is equal to 0 on Y . Moreover, when n = 2, the algebra H(Y ) of complexvalued functions on Y which can be uniformly approximated by holomorphic functions is analytic. These theorems allow us to prove that if a compact set X ⊂ C has a nontrivial Jensen measure, then X contains a nontrivial compact set Y with analytic algebra H(Y ).

Approximation of plurisubharmonic functions by multipole Green functions

For a strongly hyperconvex domain D C C n we prove that multipole pluricomplex Green functions are dense in the cone in L 1 (D) of negative plurisubharmonic functions with zero boundary values.

Bernstein–Walsh inequalities and the exponential curve in ℂ²

It is shown that for the pluripolar set K = {(z, e z ): |z| < 1} in C 2 there is a global Bernstein-Walsh inequality: If P is a polynomial of degree n on C 2 and |P| < 1 on K, this inequality gives an upper bound for |P(z, w)| which grows like exp(1/2n 2 log n). The result is used to obtain sharp estimates for |P(z, e 2 )|.

Jensen measures and analytic multifunctions

This paper describes plurisubharmonic convexity and hulls, and also analytic multifunctions in terms of Jensen measures. In particular, this allows us to get a new proof of Słodkowskis theorem stating that multifunctions are analytic if and only if their graphs are pseudoconcave. We also show that multifunctions with plurisubharmonically convex fibers are analytic if and only if their graphs locally belong to plurisubharmonic hulls of their boundaries. In the last section we prove that minimal analytic multifunctions satisfy the maximum principle and give a criterion for the existence of holomorphic selections in the graphs of analytic multifunctions.