Marius Mitrea
Department of Mathematics
University of Missouri-Columbia

Abstract: The so-called shift theorem for the Laplacian aims to identify those functional-analytic settings in which the solution of the equation Delta u=f (with homogeneous Dirichlet and Neumann boundary conditions) is two units smoother that the datum f. When the domain in which this PDE is considered is smooth, such a result is valid on all the classical smoothness spaces, of Hardy-Besov-Sobolev-Triebel-Lizorkin type, without extra restrictions on the indices involved (integrability and smoothness). On the other hand, counterexamples due to Dahlberg and Jerison-Kenig paint a dramatically different picture when the domain in question is allowed to have an irregular boundary. In this talk I will discuss some recent progress in this direction and present a complete, sharp answer to the question of the regularity of Green potentials in Lipschitz domains. As a corollary, this yields a solution of the conjecture made by Chang-Krantz-Stein in the 90's to the effect that two derivatives on the Green potential is a bounded mappings in the context of local Hardy spaces Hp for p<1 sufficiently close to 1.