On $1/2$ estimate for global Newlander-Nirenberg theorem

Abstract

Given a formally integrable almost complex structure X defined on the closure of a bounded domain D ⊂ 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 D that transform X into the standard complex structure, under certain geometric and regularity assumptions on D. In this paper we prove a regularity result of this problem. Assuming D is a strictly pseudoconvex domain in C n with C 2 boundary, and that the almost structure X is of the Hölder-Zygmund class Λ r (D) for r > 1, we prove the existence of a global diffeomorphism (independent of r) in the class Λ r+ 1 2 −ε (D), for any ε > 0. Contents 1. Introduction 2. Preliminaries 2.1. Function space and stability of constant 2.2. Smoothing operator on bounded Lipschitz domains 2.3. Paraproduct estimate 2.4. ∂ equation in Hölder space of negative smooth index 3. Norm estimates for the error term 4. Iteration Scheme and convergence of maps References