TY - JOUR
T1 - The Mutual Singularity of Harmonic Measure and Hausdorff Measure of Codimension Smaller than One
AU - Tolsa, Xavier
PY - 2021/9/1
Y1 - 2021/9/1
N2 - Let ω ℝn+1 be open and let E ω with 0 < Hs(E) < ∞, for some s ϵ (n, n + 1), satisfy a local capacity density condition. In this paper it is shown that the harmonic measure cannot be mutually absolutely continuous with the Hausdorff measure Hs on E. This answers a question of Azzam and Mourgoglou, who had proved the same result under the additional assumption that ω is a uniform domain.
AB - Let ω ℝn+1 be open and let E ω with 0 < Hs(E) < ∞, for some s ϵ (n, n + 1), satisfy a local capacity density condition. In this paper it is shown that the harmonic measure cannot be mutually absolutely continuous with the Hausdorff measure Hs on E. This answers a question of Azzam and Mourgoglou, who had proved the same result under the additional assumption that ω is a uniform domain.
UR - http://www.scopus.com/inward/record.url?scp=85122311597&partnerID=8YFLogxK
U2 - 10.1093/imrn/rnz197
DO - 10.1093/imrn/rnz197
M3 - Article
AN - SCOPUS:85122311597
SN - 1073-7928
VL - 2021
SP - 13783
EP - 13811
JO - International Mathematics Research Notices
JF - International Mathematics Research Notices
IS - 18
ER -