THE REGULARITY PROBLEM FOR ELLIPTIC OPERATORS WITH BOUNDARY DATA IN HARDY-SOBOLEV SPACE HS

Let Omega be a Lipschitz domain in R-n, n >= 3, and L = divA del be a second order elliptic operator in divergence form. We will establish that the solvability of the Dirichlet regularity problem for boundary data in Hardy-Sobolev space HS1 is equivalent to the solvability of the Dirichlet regularity problem for boundary data in H-1,H-p for some 1 < p < infinity. This is a "dual result" to a theorem in [7], where it has been shown that the solvability of the Dirichlet problem with boundary data in BMO is equivalent to the solvability for boundary data in L-p(partial derivative Omega) for some 1 < p < infinity.