On Local Behavior of Holomorphic Functions Along Complex Submanifolds of C^N

TL;DR

This paper investigates the local behavior of holomorphic functions along complex submanifolds in multi-dimensional complex space, extending Bernstein inequalities to higher dimensions and providing new insights into their properties.

Contribution

It generalizes Bernstein type inequalities from curves to higher-dimensional submanifolds, offering new theoretical results on holomorphic functions in several complex variables.

Findings

General results on local behavior of holomorphic functions along submanifolds

Multi-dimensional Bernstein inequalities for transcendental curves

Extension of classical inequalities to complex submanifolds

Abstract

In this paper we establish some general results on local behavior of holomorphic functions along complex submanifolds of $\Co^{N}$. As a corollary, we present multi-dimensional generalizations of an important result of Coman and Poletsky on Bernstein type inequalities on transcendental curves in $\Co^{2}$.