On the Hausdorff dimension of harmonic measure in higher dimension
On the Hausdorff dimension of harmonic measure in higher dimension
复制标题
高维调和测度的Hausdorff维数
DOI：
10.1007/bf01389238
复制
发表时间：
1987
影响因子：
3.1
通讯作者：
J. Bourgain
中科院分区：
文献类型：
--
作者：
J. Bourgain
设A=I~n\E是F，n中的一个域，其中E是紧集。记o~(A，A，x)是A的调和测度，在x~Rd处求值.根据S定理[O]，oE=o(A，.，x)关于d维勒贝格测度是奇异的。对于d>；2和一般结构域，这一结果似乎是迄今为止唯一已知的局部化性质。最近，P.Jones和T.Wolff[J-W]证明了对于d=2，S(O9~)的维数至多为1。这一结果完善了N.G.Makarov和L.Carleson(见I-M和[C2])的工作。设g是A~r的格林函数，极点在C*=r{O9}上。在Carleson和Jones-Wolff引理中，积分~g，~g~nnlog~nn作为边界
Assume A=I~n\E a domain in F, n, where E is a compact set. Denote o~(A,A,x) the harmonic measure for A of A, evaluated at x ~ R d. According to 0ksendal 's theorem [O], o E= o(A, . ,x) is singular with respect to d-dimensional Lebesgue measure. For d > 2 and general domains, this result seemed to be so far the only known localization property. Recently for d=2 , it has been shown by P. Jones and T. Wolff [J-W] that S(o9~) has dimension at most 1. This result completes previous work due to N.G. Makarov and L. Carleson (see I-M] and [C2]). Let g be the Green's function of A ~ r with pole at some point of C * = r {o9}. In both the Carleson and Jones-Wolff arguments, the integral ~g , ~g ~nn log ~nn as boundary