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Let $\Omega$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divtA\nabla$ be a second order elliptic operator in divergence form. We willestablish that the solvability of the Dirichlet regularity problem for boundarydata in Hardy-Sobolev space $\HS$ is equivalent to the solvability of theDirichlet regularity problem for boundary data in $H^{1,p}$ for some$1<p<\infty$. This is a "dual result" to a theorem in \cite{DKP09}, where ithas been shown that the solvability of the Dirichlet problem with boundary datain $\text{BMO}$ is equivalent to the solvability for boundary data in$L^p(\partial\Omega)$ for some $1<p<\infty$.

The regularity problem for elliptic operators with boundary data in Hardy-Sobolev space $HS^1$