On $1\mathrm{/}2$ estimate for global Newlander-Nirenberg theorem

Given a formally integrable almost complex structure $X$ defined on the closure of a bounded domain $D\subset {\mathbb{C}}^n$, and provided that $X$ is sufficiently close to the standard complex structure, the global Newlander-Nirenberg problem asks whether there exists a global diffeomorphism defined on $\bar{D}$ that transforms $X$ into the standard complex structure, under certain geometric and regularity assumptions on $D$. In this paper we prove a quantitative result of this problem. Assuming $D$ is a strictly pseudoconvex domain in ${\mathbb{C}}^n$ with $C^2$ boundary, and that the almost structure $X$ is of the Hölder-Zygmund class ${\mathrm{\Lambda }}^r(\bar{D})$ for $r>\frac{3}{2}$, we prove the existence of a global diffeomorphism (independent of $r$) in the class ${\mathrm{\Lambda }}^{r+\frac{1}{2}-\epsilon }(\bar{D})$, for any $\epsilon >0$.