论文标题
Riesz在外部Lipschitz域和应用上转换
Riesz transform on exterior Lipschitz domains and applications
论文作者
论文摘要
令$ {\ mathscr {l}} = - \ text {div} a \ nabla $是$ \ mathbb {r}^n $，$ n \ ge 2 $的均匀椭圆运算符。令$ω$为外部Lipschitz域，然后让$ {\ Mathscr {l}} _ d $和$ {\ Mathscr {l}} _ n $为操作员$ {\ Mathscr {l Mathscr {l}} $，分别是$ firichlet和neumann cormance的$上的$ω$。我们建立了Riesz的界限，转换$ \ nabla {\ Mathscr {l}} _ d^{ - 1/2} $，$ \ nabla {\ nabla {\ Mathscr {l}} _ n^n^{ - 1/2} $ in $ l^p $ space in $ l^p $ space in $ l^p $ space in $ l^p $ space。作为副产品，我们显示反向不等式$ \ | {\ Mathscr {l}} _ d^{1/2} f \ | _ {l^p（ω）} \ le c \ | \ nabla f \ | _ _ _ _ | _ {l^p（ω）} $ 1 <p <p <p <p <p <可以推广该证明，以显示Riesz变换的界限，对于外部Lipschitz或$ C^1 $域中的VMO系数的操作员。这些估计值也可以应用于不均匀的Dirichlet和Neumann问题。这些结果即使对于外部Lipschitz上Laplacian操作员的Dirichlet和Neumann和$ C^1 $域中也是新的。
Let ${\mathscr{L}}=-\text{div}A\nabla$ be a uniformly elliptic operator on $\mathbb{R}^n$, $n\ge 2$. Let $Ω$ be an exterior Lipschitz domain, and let ${\mathscr{L}}_D$ and ${\mathscr{L}}_N$ be the operator ${\mathscr{L}}$ on $Ω$ subject to the Dirichlet and Neumann boundary values, respectively. We establish the boundedness of the Riesz transforms $\nabla{\mathscr{L}}_D^{-1/2}$, $\nabla {\mathscr{L}}_N^{-1/2}$ in $L^p$ spaces. As a byproduct, we show the reverse inequality $\|{\mathscr{L}}_D^{1/2}f\|_{L^p(Ω)}\le C\|\nabla f\|_{L^p(Ω)}$ holds for any $1<p<\infty$. The proof can be generalized to show the boundedness of the Riesz transforms, for operators with VMO coefficients on exterior Lipschitz or $C^1$ domains. The estimates can be also applied to the inhomogeneous Dirichlet and Neumann problems. These results are new even for the Dirichlet and Neumann of the Laplacian operator on the exterior Lipschitz and $C^1$ domains.
