On $1/2$ 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 $\overline D$ 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 H\"older-Zygmund class $\Lambda^r(\overline D)$ for $r>\frac{3}{2}$, we prove the existence of a global diffeomorphism (independent of $r$) in the class $\Lambda^{r+\frac12-\varepsilon}(\overline D)$, for any $\varepsilon>0$.