Mathematical Sciences: Solvability and Estimates for the Tangential Cauchy-Riemann Operators

点击翻译按钮获取中文摘要

Work will be done on the solvability and estimates for the tangential Cauchy-Riemann operators. The global problem is an extension of earlier work on square- integrable existence and estimates for the boundary d- bar operator on weakly pseudo-convex boundaries. Efforts to obtain optimal Holder and power integral estimates, as well as the nonisotropic properties of the solutions will be done. The method proposed is to construct the integral kernels for the operator by fusing the boundary invariants and finite type conditions into the construction. Local solvability under condition Y(q) will also be studied using kernel methods. To obtain estimates for the local solution, work is planned on the analysis of singular integrals with singularities supported on a large set. Applications to problems of prescribing zeros of functions in the Nevanlinna class will be pursued. The generalized Blaschke condition must be satisfied on strongly pseudoconvex domains. The solution to this follows lines similar to the planned research on weakly pseudo-convex domains. Some related condition may provide the desired classification.