  We show that if $u$ is a solution to a linear elliptic differential equation
of order $2m\geq 2$ in the half-space with $t$-independent coefficients, and if
$u$ satisfies certain area integral estimates, then the Dirichlet and Neumann
boundary values of $u$ exist and lie in a Lebesgue space $L^p(\mathbb{R}^n)$ or
Sobolev space $\dot W^p_{\pm 1}(\mathbb{R}^n)$. Even in the case where $u$ is a
solution to a second order equation, our results are new for certain values
of~$p$.
