On the local solution of the tangential Cauchy-Riemann equations

Abstract

We study the solution operators P and homotopy formula introduced by G. M. Henkin for the tangential Cauchy-Riemann complex of a suitable small domain D on a strictly pseudoconvex real hypersurface in complex n-space. The main difficulties stem from the fact that P is an integral operator with a rather complicated kernel. For U ⊂⊂ D, we derive a Ck-norm estimate of the form ∥Pφ∥U, k ≦ K∥φ∥D, k, where the constant K blows up as U increases to D. We obtain careful control of the rate of this blow-up and of the dependence of K on the derivatives of the function defining the real hypersurface. Our estimates are sufficient for application to the local CR embedding problem.