The regularity problem for elliptic operators with boundary data in Hardy–Sobolev space HS

Let Ω be a Lipschitz domain in Rn,n≥3, andL=\divtA∇ be a second order elliptic operator indivergence form. We will establish that the solvability of theDirichlet regularity problem for boundary data in Hardy–Sobolevspace \HS is equivalent to the solvability of the Dirichletregularity problem for boundary data in H1,p for some1<p<∞. This is a “dual result” to a theorem in \cite{DKP09}, where it has been shown that the solvability of theDirichlet problem with boundary data in BMO is equivalentto the solvability for boundary data in Lp(∂Ω) forsome 1<p<∞.