$\epsilon$-Approximability and Quantitative Fatou Property on Lipschitz-graph domains for a class of non-harmonic functions

TL;DR

This paper investigates a class of functions on Lipschitz-graph domains satisfying a differential-oscillation condition, proving their $ ext{epsilon}$-approximability and establishing a quantitative Fatou theorem applicable to both harmonic and non-harmonic functions.

Contribution

It introduces a new class of functions satisfying a differential-oscillation condition and proves their $ ext{epsilon}$-approximability along with a quantitative Fatou theorem, extending previous results to non-harmonic functions.

Findings

Functions satisfying the differential-oscillation condition are $ ext{epsilon}$-approximable.

The quantitative Fatou theorem is established for this class of functions.

Includes both harmonic and non-harmonic functions such as subharmonic functions.

Abstract

We study the class of functions on Lipschitz-graph domains satisfying a differential-oscillation condition and show that such functions are $\epsilon$-approximable. As a consequence we obtain the quantitative Fatou theorem in the spirit of works e.g. by Garnett and Bortz-Hofmann. Such a class contains harmonic functions, as well as non-harmonic ones, for example nonnegative subharmonic functions, as illustrated by our discussion.