Dahlberg's theorem in higher co-dimension
Dahlberg's theorem in higher co-dimension
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DOI：
10.1016/j.jfa.2019.02.006
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发表时间：
2019-05-01
影响因子：
1.7
通讯作者：
Mayboroda, Svitlana
中科院分区：
文献类型：
--
作者：
David, Guy;Feneuil, Joseph;Mayboroda, Svitlana
1977年，B.Dahlberg的著名定理证明了R-n中n-1维Lipschitz图上的调和测度关于Hausdorff测度是绝对连续的，后来这一结果被推广到更一般的非切可达区域上.本文证明了在R-n，d<；n-1中d维Lipschitz图Gamma上的Dahlberg定理在高维上的第一个类似的结果.我们构造了一个线性退化椭圆算子L，使得相应的调和测度omega(L)关于Gamma上的Hausdorff测度是绝对连续的。更一般地，我们给出了关于L系数矩阵的充分条件，这些条件保证了Omega(L)与Hausdorff测度的绝对连续性。(C)2019 Elsevier Inc.保留所有权利。
In 1977 the celebrated theorem of B. Dahlberg established that the harmonic measure is absolutely continuous with respect to the Hausdorff measure on a Lipschitz graph of dimension n - 1 in R-n, and later this result has been extended to more general non-tangentially accessible domains and beyond.In the present paper we prove the first analogue of Dahlberg's theorem in higher co-dimension, on a Lipschitz graph Gamma of dimension d in R-n, d < n -1, with a small Lipschitz constant. We construct a linear degenerate elliptic operator L such that the corresponding harmonic measure omega(L) is absolutely continuous with respect to the Hausdorff measure on Gamma. More generally, we provide sufficient conditions on the matrix of coefficients of L which guarantee the mutual absolute continuity of omega(L) and the Hausdorff measure. (C) 2019 Elsevier Inc. All rights reserved.