Sharp Hölder estimates for {̄∂} on ellipsoids and their complements via order of contact

Abstract

We generalize the results of Range and Diederich et al., finding Hölder estimates for the solution of the Cauchy-Riemann equations for higher order forms on ellipsoids. We prove a dual result near the concave boundaries of complemented complex ellipsoids. In all cases the Hölder exponents are characterized in terms of the order of contact of the boundary of the domain with complex linear spaces of the appropriate dimension. Optimality is demonstrated in the convex settings, and for ( 0 , 1 ) (0,1) forms in the concave setting. Partial results are given for complemented real ellipsoids and a method for demonstrating optimality of Hortmann’s result on complemented strictly pseudoconvex domains is given for ( 0 , 1 ) (0,1) forms.