On boundary properties of asymptotically holomorphic functions

It is well known that for a generic almost complex structure on an almost complex manifold $(M,J)$ all holomorphic (even locally) functions are constants. For this reason the analysis on almost complex manifolds concerns the classes of functions which satisfy the Cauchy-Riemann equations only approximately. The choice of such a condition depends on a considered problem. For example, in the study of zero sets of functions the quasiconformal type conditions are very natural. This was confirmed by the famous work of S. Donaldson. In order to study the boundary properties of classes of functions (on a manifold with boundary) other type of conditions are suitable. In the present paper we prove a Fatou type theorem for bounded functions with $\overline\partial_J$ differential of a controled growth on smoothly bounded domains in an almost complex manifold. The obtained result is new even in the case of $\mathbb{C}^n$ with the standard complex structure. Furthermore, in the case of $\mathbb{C}^n$ we obtain results with optimal regularity assumptions. This generalizes several known results.