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

Abstract

We consider divergence form elliptic operators $L=-\dv A(x)\nabla$, defined in ${\mathbb{R}}^{n+1}=\{(x,t)\in {\mathbb{R}}^n\times \mathbb{R}\},n\ge 2$, 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 $\mathbf{v}\in W_{loc}^{1,2},$ we have the bilinear estimate $$|{\iint }_{{\mathbb{R}}_{+}^{n+1}}\nabla u\cdot \bar{\mathbf{v}}dxdt|\le C\underset{t>0}{\sup }\Vert u(\cdot ,t){\Vert }_{L^2({\mathbb{R}}^n)}(\Vert |t\nabla \mathbf{v}\Vert |+\Vert N_{\ast }\mathbf{v}{\Vert }_{L^2({\mathbb{R}}^n)}),$$ where $\Vert |F\Vert |\equiv ({\iint }_{{\mathbb{R}}_{+}^{n+1}}|F(x,t){|}^2{t}^{-1}dxdt{)}^{1/2},$ and where $N_{\ast }$ 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.