Maximal Estimates in Several Complex Variables

Complex analysis in one variable is closely tied to the study of harmonic functions. In several complex variables, there is also a second-order PDE that is fundamental to the study of the holomorphic functions: the $\bar\partial$-Neumann problem. In contrast with the one variable case, the boundary condition for the $\bar\partial$-Neumann problem is non-coercive, so solution operators for the $\bar\partial$-Neumann problem gain at most one derivative in the Sobolev scale. Given this constraint, we say that a domain admits maximal estimates if the solution operator for the $\bar\partial$-Neumann problem gains two derivatives in every direction except one. We will see that a large class of domains admit maximal estimates, and many difficult problems in several complex variables are easier to study on such domains.