  Let $\Omega$ be a Lipschitz domain in $\mathbb R^n,n\geq 3,$ and $L=\divt
A\nabla$ 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 $\HS$ is equivalent to the solvability of the
Dirichlet 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 it
has been shown that the solvability of the Dirichlet problem with boundary data
in $\text{BMO}$ is equivalent to the solvability for boundary data in
$L^p(\partial\Omega)$ for some $1<p<\infty$.
