Shcherbina's Theorem for Finely Holomorphic Functions

TL;DR

This paper extends classical theorems in complex analysis to C-1 functions and finely holomorphic functions, establishing conditions under which the graph of such functions is pluripolar and relates to the Cauchy-Riemann equations.

Contribution

It proves a version of Shcherbina's theorem for C-1 functions and finely holomorphic functions, linking pluripolarity of graphs to the Cauchy-Riemann equations.

Findings

The graph of a C-1 function is pluripolar only if it satisfies the Cauchy-Riemann equations.

An analogue of Sadullaev's theorem is established for manifolds of class C-1.

The results connect pluripolarity with fine topology and complex differentiability.

Abstract

We prove an analogue of Sadullaev's theorem concerning the size of the set where a maximal totally real manifold can meet a pluripolar set. The manifold has to be of class C-1 only. This readily leads to a version of Shcherbina's theorem for C-1 functions f that are defined in a neighborhood of certain compact sets K in the complex plane. If the graph of f on K is pluripolar, then f satisfies the Cauchy Riemann equations in the closure of the fine interior of K.