ɛ-approximability and quantitative Fatou property on Lipschitz-graph domains for a class of non-harmonic functions

We study the class of functions on Lipschitz-graph domains satisfying a differential-oscillation condition and show that such functions are $\varepsilon$ -approximable. As a consequence, we obtain the quantitative Fatou theorem in the spirit of works, for example, by Garnett [6] and Bortz–Hofmann [1]. Such a class contains harmonic functions, as well as non-harmonic ones, for example, nonnegative subharmonic functions whose gradient norm is quasi-nearly subharmonic, as illustrated by our discussion.