  We consider divergence form elliptic operators $L=-\dv A(x)\nabla$, defined
in $\mathbb{R}^{n+1}=\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}\}, n \geq 2$,
where the $L^{\infty}$ coefficient matrix $A$ is $(n+1)\times(n+1)$, uniformly
elliptic, complex and $t$-independent. Using recently obtained results
concerning the boundedness and invertibility of layer potentials associated to
such operators, we show that if $Lu=0$ in $\mathbb{R}^{n+1}_+$, then for any
vector-valued ${\bf v} \in W^{1,2}_{loc},$ we have the bilinear estimate
$$|\iint_{\mathbb{R}^{n+1}_+} \nabla u \cdot \bar{{\bf v}} dx dt |\leq
C\sup_{t>0} \|u(\cdot,t)\|_{L^2(\mathbb{R}^n)}(\||t \nabla {\bf v}\|| +
\|N_*{\bf v}\|_{L^2(\mathbb{R}^n)}),$$ where $\||F\|| \equiv
(\iint_{\mathbb{R}^{n+1}_+} |F(x,t)|^2 t^{-1} dx dt)^{1/2},$ and where $N_*$ is
the usual non-tangential maximal operator. The result is new even in the case
of real symmetric coefficients, and generalizes the analogous result of
Dahlberg for harmonic functions on Lipschitz graph domains.
