A Newlander-Nirenberg theorem for manifolds with boundary (Q1822775)

Let \(M\) be a smooth manifold of real dimension \(2n\) with smooth boundary \(bM\), and let \(U\) be a neighbourhood in the relative topology of \(M\) of a given point \(z_ 0\in bM\). We say that an almost complex structure is defined in \(U\) if there exists a subbundle \(L\) of fiber dimension \(n\) of the complexified tangent bundle \(CT(M)\) such that for each \(z\in U\), \(L_ z\cap \bar L_ z=0\). This structure is called integrable if \(L\) is closed under brackets. The author proves that the analog of the Newlander-Nirenberg theorem holds if the boundary of \(M\) is pseudoconvex near \(z_ 0\).