Mathematics

Dr. Debraj Chakrabarti
Central Michigan University

One of the fundamental problems of several complex variables is to construct holomorphic functions and other analytic objects with prescribed properties. Often, this can be reduced to solving the system of inhomogeneous Cauchy-Riemann equations with estimates in a particular norm – known as the \\overline{\\partial}-problem . In 1965, Hörmander showed that in bounded pseudoconvex domains, the \\overline{\\partial}-problem can be solved with L^2estimates.

We consider the question of solving the \\overline{\\partial}-problem in the annulus bounded by two pseudoconvex domains, where the “hole” is allowed to be nonsmooth. We obtain estimates for this problem in L^2 spaces using a gluing technique, and prove vanishing results for the L^2− \\overline{\\partial}-cohomology.

This is joint work with Mei-Chi Shaw (Notre Dame) and Christine Laurent-Thiébaut (Grenoble).