Geometry of Strictly Convex Domains and an Application to the Uniform Estimate of the $\overline\partial$-Problem

Abstract

Abstract.In this paper, we construct a nice defining function p for a bounded smooth strictly convex domain ÎÎ in R" with explicit gradient and Hessian estimates near the boundary 9Í2 of Í2 . From the approach, we deduce that any two normals through d SI do not intersect in any tubular neighborhood of dSl with radius which is less than ^ , where K is the maximum principal curvature of ÔÎ2 . Finally, we apply such p to obtain an explicit upper bound of the constant Cq in the Henkin's estimate ||-Hn/||/_°°(£î) < Qill/llL^tn) of the d-problem on strictly convex domains Í2 in C" . In fact, it is well known that (0.1) can always be solved for any pseudoconvex domain Q in C" without any boundary restrictions on Q [7, p. 87]. Therefore, we may restrict ourself to the category of pseudoconvex domains Q, and ask further: (0.2) If / 6 L[0 1}(Q), i.e., a (0, l)-form with coefficients in LP(Çi), and df = 0, where 1 < p < oo, then can one solve (0.1) with the following inequality:IMIlo(îî) ^ Qîll/llz/(n),