Regularity of the tangential Cauchy-Riemann complex and applications

Abstract

In a series of papers Webster ([26], [27], [28]) has shown how C-estimates for the tangential Cauchy-Riemann complex can be applied to several non-linear problems in Complex Analysis. For example, he gave a simplification of the proof of Kuranishi’s embedding theorem and an application to the integrability problem for almost CR vector bundles. Connected with Webster’s approach are some regularity assertions, whose parameters follow from the C-estimates. Therefore, an improvement of these estimates would lead to an improvement in the applications. Now the author of this article, together with Lan Ma, has given in [12] such an improvement. In two subsequent papers this has been applied to Webster’s approach. The results are formulated in two theorems at the end of this introduction. Here we want to briefly describe the estimates for the tangential CR equations. Let G be a strictly pseudoconvex domain in C with C-smooth boundary and 0 ∈ bG. By a local homotopy formula for the tangential Cauchy-Riemann operator ∂b we mean the following. There exists a neighborhood base {M} of relatively open sets M , with 0 ∈ M ⊂ bG, and on each M there are given operators Rq (q = 1, 2, . . . , n− 2), which fullfill the equation