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


Author: Steve Hofmann
Journal: Proc. Amer. Math. Soc. 136 (2008), 4223-4233
MSC (2000): Primary 42B20, 42B25, 35J25
DOI: https://doi.org/10.1090/S0002-9939-08-09500-2
Published electronically: July 25, 2008
MathSciNet review: 2431035
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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 $ {\bf v} \in W^{1,2}_{loc},$ we have the bilinear estimate

$\displaystyle \left\vert\iint_{\mathbb{R}^{n+1}_+} \nabla u \cdot \overline{{\... ...t \nabla {\bf v}\Vert\vert + \Vert N_*{\bf v}\Vert _{L^2(\mathbb{R}^n)}\right),$

where $ \Vert\vert F\Vert\vert \equiv \left(\iint_{\mathbb{R}^{n+1}_+} \vert F(x,t)\vert^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}_+.$


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Additional Information

Steve Hofmann
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: hofmann@math.missouri.edu

DOI: https://doi.org/10.1090/S0002-9939-08-09500-2
Received by editor(s): April 27, 2007
Published electronically: July 25, 2008
Additional Notes: The author was supported by the National Science Foundation
Communicated by: Michael T. Lacey
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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