Title: Convolutions, Singular Integrals, and Several Complex Variables
Abstract: The Dirichlet problem for the unit disk is to find a harmonic function with certain prescribed boundary values on the unit circle. The solution is given by taking the convolution of the boundary value function with the Poisson kernel. The theory of such convolution operators is quite well understood. The theory is more subtle when we generalize to consider integral operators that have non-integrable kernels or are not of convolution type. In this talk we give an overview of this theory and discuss its connections with current research in several complex variables.