  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 $\varepsilon>0$.
