Dahlberg's bilinear estimate for solutions of divergence form complex elliptic equations
| dc.creator | Hofmann, S. | |
| dc.date | 2007-05-07 | |
| dc.date.accessioned | 2026-07-07T07:59:44Z | |
| dc.date.available | 2026-07-07T07:59:44Z | |
| dc.description | 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. | |
| dc.identifier | https://arxiv.org/abs/0705.0839 | |
| dc.identifier | http://arxiv.org/abs/0705.0839 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/128482 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.title | Dahlberg's bilinear estimate for solutions of divergence form complex elliptic equations | |
| dc.type | text |