Uniform estimates for the $$\bar \partial $$ -equation on domains with piecewise smooth strictly pseudoconvex boundaries

Abstract

Henkin [3] and Grauert-Lieb I7] pioneered in the investigation of uniform estimates for the ~-equation. They independently proved the following: if D is a strictly pseudoconvex domain in fEn with smooth boundary and i f f is a uniformly bounded R-closed C °~ (0, 1)-form on D, then there exists a uniformly bounded C ° function u on D with du = f. Their proofs depend on the explicit construction of a solution by means of holomorphic kernels introduced by Henkin [2] and Ramirez [19]. Later, by using a local version of the Grauert-Lieb method, Kerzman 1'9] generalized the result to a strictly pseudoconvex domain with smooth boundary in a Stein manifold. Moreover, he showed that the type of solution constructed by Grauert-Lieb for such a domain yields also LP-estimates and H61der estimates with exponents 1/2. By modifying Henkin's solution of the ~-equation, Henkin and Romanov 1'5] obtained the H61der estimate with exponent t/2. Making use of Koppelman's results 1'1 t], Lieb [13, 14] proved that there exist uniform estimates and H/ilder estimates with exponent < 1/2 for the ~-equation for (0, q)-forms on strictly pseudoconvex domains with smooth boundary. Independently ~vrelid [16] also obtained uniform estimates and L p estimates for the ~-equation for (0, q)-forms on the same class of domains. Recently Henkin 1'4] announced the solution of the R-equation with uniform bounds for (0, 1)-forms for certain analytic polyhedra and indicated that his proof makes use of the type of Cauchy-Fantappie kernel introduced by Norguet 1'15]. In this paper we investigate uniform estimates for solutions of the R-equation on a domain with piecewise smooth strictly pseudoconvex boundary. More precisely, suppose D is a bounded domain in C n with aD covered by finitely many open subsets Uj(l _~j_~ k) ofC n and suppose 01 is a ¢2 strictly plurisubharmonic function on UI(1 < j < k) such that