SOLVING THE d AND ∂-EQUATIONS IN THIN TUBES AND APPLICATIONS TO MAPPINGS

We construct a family of integral kernels for solving the ∂-equation with Ck and Holder estimates in thin tubes around totally real submanifolds in Cn (theorems 1.1 and 3.1). Combining this with the proof of a theorem of Serre we solve the d-equation with estimates for holomorphic forms in such tubes (theorem 5.1). We apply these techniques and a method of Moser to approximate Ck-diffeomorphisms between totally real subman- ifolds in Cn in the Ck-topology by biholomorphic mappings in tubes, by unimodular and symplectic biholomorphic mappings, and by automorphisms of Cn. &1. The results. Let Cn denote the complex n-dimensional Euclidean space with complex coordinates z = (z1, . . . , zn). A compact Ck-submanifold M ⊂ Cn (k ≥ 1), with or without boundary, is totally real if for each z ∈ M the tangent space TzM (which is a real subspace of TzCn) contains no complex line; equivalently, the complex subspace T C z M = TzM + iTzM of TzCn has complex dimension m = dim RM for each z ∈ M. We denote by TδM = {z ∈ Cn: dM(z) < δ} the tube of radius δ > 0 around M; here |z| is the Euclidean norm of z ∈ Cn and dM(z) = inf{|z − w|: w ∈ M }. For any open set U ⊂ Cn and integers p, q ∈ Z+ we denote by Cl p,q(U) the space of differential forms of class Cl and of bidegree (p, q) on U. For each multiindex α ∈ Z2n + we denote by ∂α the corresponding partial derivative of order |α| with respect to the underlying real coordinates on Cn.