论文标题
椭圆度度量和平方函数估计1侧和弦域域
Elliptic measures and Square function estimates on 1-sided chord-arc domains
论文作者
论文摘要
在诸如Lipschitz或Chord-arc域之类的良好环境中，众所周知，对于某些有限的$ p $，Dirichlet问题对椭圆运算符的Dirichlet问题的解决性等同于以下事实。反过来，当且仅当每个有界空溶液的梯度满足CARLEON测量估计值时，这些条件中的任何一个都会发生。最近已将其扩展到更粗糙的设置，例如1侧和弦域域的设置，即定量打开并与Ahlfors-David常规的边界连接的集合。 在本文中，我们在相同的环境中工作，并考虑了后者等效性的定性类似物，表明人们可以根据截断的圆锥形正方形函数在任何有限的null null溶液中都以有限性来表征表面度量相对于椭圆度度量的绝对连续性。由于我们的主要结果专门针对拉普拉斯操作员和一些先前的结果，我们表明域的边界在且仅当截断的锥形平方函数几乎在任何有限的谐波函数的地方都是有限的。另外，我们获得了两个给定的椭圆运算符$ l_1 $和$ l_2 $，相对于$ l_1 $的椭圆度度量的绝对连续性与同一属性相当于$ l_2 $的同一属性，前提是该系数的不同意是在截断的锥度中的某些近处估计值，几乎在任何地方均可估计。最后，对于$ l_2 $是$ l_1 $的转孔或其对称零件的情况，我们表明了相应的绝对连续性的等效性，假设系数的抗对称部分几乎在截短的圆锥上具有一定的控制振荡。
In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $L^p$, for some finite $p$, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $A_\infty$. In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundary which is Ahlfors-David regular. In this paper, we work in the same environment and consider a qualitative analog of the latter equivalence showing that one can characterize the absolute continuity of the surface measure with respect to the elliptic measure in terms of the finiteness almost everywhere of the truncated conical square function for any bounded null solution. As a consequence of our main result particularized to the Laplace operator and some previous results, we show that the boundary of the domain is rectifiable if and only if the truncated conical square function is finite almost everywhere for any bounded harmonic function. Also, we obtain that for two given elliptic operators $L_1$ and $L_2$, the absolute continuity of the surface measure with respect to the elliptic measure of $L_1$ is equivalent to the same property for $L_2$ provided the disagreement of the coefficients satisfy some quadratic estimate in truncated cones for almost everywhere vertex. Finally for the case on which $L_2$ is either the transpose of $L_1$ or its symmetric part we show the equivalence of the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for almost every vertex.
