散度型复椭圆型方程解的Dahlberg双线性估计
Dahlberg's bilinear estimate for solutions of divergence form complex elliptic equations
摘要：
考虑散度型椭圆算子$L=-\dv A(x)\nabla$，定义在$\mathbb{R}^{n+1}=\{(x,t)\in\mathbb{R}^{n}\times\mathbb{R}\}, n \geq 2$中，其中$L^{\infty}$系数矩阵$A$是$(n+1)\times(n+1)$，一致椭圆的，复的和$t$无关的.使用最近获得的结果有关的有界性和可逆性层潜在的这些运营商，我们表明，如果$Lu=0$在$\mathbb{R}^{n+1}_+$，那么对于任何向量值${\bf v} \in W^{1,2}_{loc},$，我们有双线性估计$$|\iint_{\mathbb{R}^{n+1}_+} \nabla u \cdot \bar{\bf v} dx dt |\leq C\sup_{t>0} \|u(\cdot,t)\|_{L^2(\mathbb{R}^n)}(\||t \nabla {\bf v}\|| + \|N_*{\bf v}\|_{L^2(\mathbb{R}^n)}),$$其中$\||F\|| \equiv (\iint_{\mathbb{R}^{n+1}_+} |F(x,t)|^2 t^{-1} dx dt)^{1/2},$和$N_*$是通常的非切最大运营商。这个结果在实对称系数的情况下也是新的，推广了Dahlberg关于Lipschitz图域上调和函数的类似结果.
英文摘要：
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.
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