On Local Behavior of Holomorphic Functions Along Complex Submanifolds of C

In this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of CN . As a corollary, we present multi-dimensional generalizations of an important result of Coman and Poletsky on Bernstein type inequalities on transcendental curves in C2. 1. Formulation of Main Results 1.1. In this paper we establish some general results on restrictions of holomorphic functions to complex submanifolds of C . The subject pertains to the area of the, so-called, polynomial inequalities for analytic and plurisubharmonic functions that includes, in particular, Bernstein, Markov and Remez type inequalities. Recently there has been a considerable interest in such inequalities in connection with various problems of analysis. Let us recall that the classical univariate inequalities for polynomials have appeared in approximation theory and for a long time have been considered as technical tools for proofs of Bernstein type inverse theorems. At the present time polynomial type inequalities have been found a lot of important applications in areas which are well apart from approximation theory. We will only briefly mention several of these areas. The papers [GM], [Bou] and [KLS] apply polynomial inequalities with different integral norms to study some problems of Convex Geometry (in particular, the famous Slice Problem). In the papers [B1], [B2], [BB], [G], [P] and [PP] and books [DS] and [JW] Chebyshev-Bernstein and related Markov type inequalities are used to explore a wide range of properties of the classical spaces of smooth functions including Sobolev type embeddings and trace theorems, extensions and differentiability. ∗Research supported in part by NSERC. 2000 Mathematics Subject Classification. Primary 32A17, Secondary 46E15.