## Elliptic Problems and Singular Integrals on Besov-Triebel-Lizorkin Spaces

- April 2, 2002
- 3:30 p.m.
- LeConte 312

## Abstract

The Besov and Triebel-Lizorkin classes are smoothness scales which encompass in a unified fashion many of the classical function spaces, such as Lebesgue, Sobolev, Holder, Hardy, BMO, Zygmund, etc.

The aim of this talk is to explore the correlation between the smoothness of solutions and that of data (measured on Besov and Triebel-Lizorkin scales) for elliptic PDE's in nonsmooth domains. In contrast to the corresponding situation for the smooth setting, there are necessary limitations of the theory, caused by the `worst' type of (boundary) irregularity allowed. Specific examples to be considered include Poisson type problems, complex powers of elliptic operators, and Hodge decompositions in domains which can be described locally by means of graphs of Lipschitz functions. Our approach emphasizes the systematic use of singular integral operators of Calderon-Zygmund type and all our results are sharp in this class of domains.