Dahlberg’s bilinear estimate for solutions of divergence form complex elliptic equations

Abstract:

We consider divergence form elliptic operators $L=-\operatorname {div} A(x)\nabla$, 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 $\textbf {v} \in W^{1,2}_{loc},$ we have the bilinear estimate \[ \left |\iint _{\mathbb {R}^{n+1}_+} \nabla u \cdot \overline {\textbf {v}} dx dt \right |\leq C\sup _{t>0} \|u(\cdot ,t)\|_{L^2(\mathbb {R}^n)}\left ( \||t \nabla \textbf {v}\|| + \|N_*\textbf {v}\|_{L^2(\mathbb {R}^n)}\right ),\] where $\||F\|| \equiv \left (\iint _{\mathbb {R}^{n+1}_+} |F(x,t)|^2 t^{-1} dx dt\right )^{1/2},$ and where $N_*$ is the usual non-tangential maximal operator. The result is new even in the case of real symmetric coefficients and generalizes an analogous result of Dahlberg for harmonic functions on Lipschitz graph domains. We also identify the domain of the generator of the Poisson semigroup for the equation $Lu=0$ in $\mathbb {R}^{n+1}_+.$