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

Authors:	Steve Hofmann

Affiliation:	Department of Mathematics, University of Missouri, Columbia, Missouri 65211

Abstract:	We consider divergence form elliptic operators , defined in $mathbb{R}^{n+1}={(x,t)inmathbb{R}^{n}timesmathbb{R}},, n geq 2$ , where the $L^{infty}$ coefficient matrix is , uniformly elliptic, complex and -independent. Using recently obtained results concerning the boundedness and invertibility of layer potentials associated to such operators, we show that if in $mathbb{R}^{n+1}_+$ , then for any vector-valued ${bf v} in W^{1,2}_{loc},$ we have the bilinear estimate $displaystyle leftvertiint_{mathbb{R}^{n+1}_+} nabla u cdot overline{{... ...t nabla {bf v}Vertvert + Vert N_*{bf v}Vert _{L^2(mathbb{R}^n)}right),$ where $Vertvert FVertvert equiv left(iint_{mathbb{R}^{n+1}_+} vert F(x,t)vert^2 t^{-1} dx dtright)^{1/2},$ and where 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 in $mathbb{R}^{n+1}_+.$

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