Given a formally integrable almost complex structure J defined on the closure of a bounded domain \(D \subset \mathbb C^n\) , and provided that J 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 J 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 complex structure J belongs to the Hölder–Zygmund class \(\Lambda ^r(\overline{D})\) for \(r>\frac{3}{2}\) , we show the existence of a global diffeomorphism (independent of r) in the class \(\Lambda ^{r+\frac{1}{2}-\varepsilon }(\overline{D})\) , for any \(\varepsilon >0\) .
