| Abstract: |
| The goal of this talk to present recent developments in boundary value problems for weakly elliptic, second-order systems with constant coefficients. Specifically, for such systems we consider the higher-order regularity problem in the upper-half space when the boundary datum is arbitrarily prescribed from a Generalized Banach Function Space. A Generalized Banach Function Space is a more inclusive version of the usual notion of a Banach Function Space, which admit various function spaces, e.g. the class of Muckenhoupt weighted Morrey spaces and their preduals, a.k.a, Block spaces, that traditional Banach function spaces fail to include. We are able to successfully demonstrate well-posedness in this regime for an arbitrary amount of smoothness at the boundary by working with a general notion of a Poisson kernel, as well as a Calder\`{o}n-Zygmund theory, tailored for Generalized Banach Function Spaces. This is joint work with Marius Mitrea. |
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