论文标题
绿色功能和光滑距离
Green functions and smooth distances
论文作者
论文摘要
在本文中，我们表明，对于一类最佳的椭圆运算符，在1面和弦域上具有非平滑系数g（x）} {g（x）} - \ frac {\ nabla d（x）} {d（x）} \ big |^2d（x）dx $是carleson量度的密度，其中$ d $是适应域边界的正则距离。我们证明中的主要成分是电晕分解，与Tolsa的$α$ number兼容均匀整洁的集合。我们认为，该方法可以应用于PDE和几何测量理论的交集中的许多其他问题，尤其是我们能够将经典F.和M. Riesz Theorem的概括推导为上述椭圆运算符。
In the present paper, we show that for an optimal class of elliptic operators with non-smooth coefficients on a 1-sided Chord-Arc domain, the boundary of the domain is uniformly rectifiable if and only if the Green function $G$ behaves like a distance function to the boundary, in the sense that $\Big|\frac{\nabla G(X)}{G(X)}-\frac{\nabla D(X)}{D(X)}\Big|^2D(X) dX$ is the density of a Carleson measure, where $D$ is a regularized distance adapted to the boundary of the domain. The main ingredient in our proof is a corona decomposition that is compatible with Tolsa's $α$-number of uniformly rectifiable sets. We believe that the method can be applied to many other problems at the intersection of PDE and geometric measure theory, and in particular, we are able to derive a generalization of the classical F. and M. Riesz theorem to the same class of elliptic operators as above.
