17 Novembre 2021 17:00 CET Zoom online
Phillip S. Harrington University of Arkansas

Maximal estimates in several complex variables

Abstract 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.

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