Hölderabschätzungen für Ableitungen von Lösungen der Gleichung$\bar \partial$u = f bei streng pseudokonvexem Rand

Abstract

Using the local Kerzman kernel we prove regularity of solutions of$\bar \partial$u=f, where f is a$\bar \partial$-closed (0,1)-form in a strongly pseudoconvex domain G in ℂ^N. If f is in H^m,∞, then the solution is in$\tilde C^{m,\mu }$ forμ<1, that is, the m-th derivatives are in C^o,μ/2 and in addition areμ-Hölder continuous on curves “parallel” to the holomorphic tangent bundle$\tilde T$∂G. If f is in C^m,α with o<α<1, then the solution is in$\tilde C^{m,1 + \mu }$ forμ<α, that is, the m-th derivatives are in C^o,(1+μ/2, and they have first derivatives “parallel” to$\tilde T$∂G lying in$\tilde C^{o,\mu }$. We derive the same results for the global solution constructed by Grauert and Lieb, and similar estimates on complex manifolds.