Projection Theorems for Harmonic Measure in NTA Domains

Suppose D is an NTA domain, E D is any closed set, and P ^x ₀(E) is the projection with respect to a point x ₀D of the set E onto the boundary of D. The projection P ^x ₀ satisfies certain geometric properties so that it is a generalization of the notion of radial projection with respect to a point x ₀ onto a boundary of a domain. It is shown that the harmonic measure of E with respect to the domain DE evaluated at the point x ₀ is bounded below by a constant times the harmonic measure of the set P ^x ₀(E) with respect to the domain D evaluated at the point x ₀. The constant is independent of the set E but it may depend upon x ₀.     

BMO Solvability and Absolute Continuity of Harmonic Measure

We show that for a uniformly elliptic divergence form operator L, defined in an open set \(\Omega \) with Ahlfors–David regular boundary, BMO solvability implies scale-invariant quantitative absolute continuity (the weak-\(A_\infty \) property) of elliptic-harmonic measure with respect to surface measure on \(\partial \Omega \). We do not impose any connectivity hypothesis, qualitative, or quantitative; in particular, we do not assume the Harnack Chain condition, even within individual connected components of \(\Omega \). In this generality, our results are new even for the Laplacian. Moreover, we obtain a partial converse, assuming in addition that \(\Omega \) satisfies an interior Corkscrew condition, in the special case that L is the Laplacian.     

Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability

We show that, for disjoint domains in the euclidean space whose boundaries satisfy a nondegeneracy condition, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries. © 2017 Wiley Periodicals, Inc.     

On the Continuity and Absolute Continuity of Random Closed Sets

Abstract

In the case of real-valued random variables, the concept of absolute continuity is well-defined in terms of the absolute continuity of the probability law of a random variable with respect to the usual Lebesgue measure, since both are acting on the same Borel sigma algebra on the real line. Naturally, the same extends to random vectors with real components. A satisfactory and commonly accepted definition of absolute continuity of random closed sets is not available, while in various applications this would help in clarifying the kind of randomness of a random set. We introduce here a definition that is shown to be an extension of the concept related to real-valued random variables, such that also for random sets it is true that absolute continuity implies continuity. Significant examples and counter examples are presented to illustrate the role of our definition in concrete cases. The relationship between our definition and others in well-accepted literature is shown.     

自相似集的Hausdorff测度与连续性

对集合F Rn,以dim F和Hdim F(F)分别表示F的Hausdorff维数和dim F维Hausdorff测度.设T=T(f1,...,fm)为Rn中的自相似集,即由相似压缩组成的迭代函数系统{f1...,fm)的吸引子.假如fi(T)∩fj(T)= (i≠j),那么,对任意ε>0,存在δ>0,若D=D(g1,...,gm)为Rn中的自相似集并且sup{||fk(x)-gk(x)||:||x||≤1,1≤k≤m}<δ,则1HdimT(T)-Hdim D(D)|<ε.     

Hausdorff Dimension of Harmonic Measure for Self-Conformal Sets

Under some technical assumptions it is shown that the Hausdorff dimension of the harmonic measure on the limit set of a conformal infinite iterated function system is strictly less than the Hausdorff dimension of the limit set itself if the limit set is contained in a real-analytic curve, if the iterated function system consists of similarities only, or if this system is irregular. As a consequence of this general result the same statement is proven for hyperbolic and parabolic Julia sets, finite parabolic iterated function systems and generalized polynomial-like mappings. Also sufficient conditions are provided for a limit set to be uniformly perfect and for the harmonic measure to have the Hausdorff dimension less than 1. Some results in the spirit of Przytycki et al. (Ann. of Math.130 (1989), 1-40; Stud. Math.97 (1991), 189-225) are obtained.     

Quantitative Absolute Continuity of Harmonic Measure and the Dirichlet Problem: A Survey of Recent Progress

It is a well-known folklore result that quantitative, scale invariant absolute continuity (more precisely, the weak-A_∞ property) of harmonic measure with respect to surface measure, on the boundary of an open set Ω ⊂ Rⁿ⁺¹ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in Ω, with data in L^p(∂Ω) for some p < ∞. Drawing an analogy to the famous Wiener criterion, which characterizes the domains in which the classical Dirichlet problem, with continuous boundary data, can be solved, one may seek to characterize the open sets for which L^p solvability holds, thus allowing for singular boundary data. It has been known for some time that absolute continuity of harmonic measure is closely tied to rectifiability properties of ∂Ω, but also that rectifiability alone is not sufficient to guarantee absolute continuity. In this note, we survey recent progress in this area, culminating in a geometric characterization of the weak-A_∞ property, and hence of solvability of the L^p Dirichlet problem for some finite p. This characterization, obtained under rather optimal background hypotheses, follows from a combination of the present author's joint work with Martell, and the work of Azzam, Mourgoglou and Tolsa.     

带交互作用测度值分枝过程的绝对连续性

本文研究Meleard-Roelly（1992，1993）和Metivier（1987）构造的带交互作用的测度值分枝过程的状态性质．我们证明在自然假设下该过程关于Lebesgue测度是绝对连续的，其密度有连续修正且满足一个随机偏微分方程．     

On Domains of Attraction of Multivariate Extreme Value Distributions under Absolute Continuity

The paper gives sufficient conditions for domains of attraction of multivariate extreme value distributions. Under the assumption of absolute continuity of a multivariate distribution, the criteria enable one to examine, by using limits of some rescaled conditional densities, whether the distribution belongs to the domain of attraction of some multivariate extreme value distribution. If this is the case, the criteria also determine how to construct such an extreme value distribution. Unlike the criterion given by de Haan and Resnick [1987,Stochastic Process. Appl.2583–93], the criteria are easily applicable even when the marginal tails are not Pareto-like.     

Absolute Continuity in Periodic Waveguides

We study second order elliptic operators with periodic coefficientsin two-dimensional simply connected periodic waveguides withthe Dirichlet or Neumann boundary conditions. It is proved thatunder some mild smoothness restrictions on the coefficients,such operators have purely absolutely continuous spectra. Theproof follows a method suggested previously by A. Morame totackle periodic operators with variable coefficients in dimension2. 2000 Mathematical Subject Classification: 35J10, 35P05, 35J25.     

Continuity of Harmonic Functions for Non-local Markov Generators

We consider a class of pure jump Markov processes in ${\mathbb R}^d$ whose jump kernels are comparable to those of symmetric stable processes. We prove a support theorem, a lower bound on the occupation times of sets, and show that we can approximate resolvents using smooth functions.     

关于Fuzzy测度的绝对连续性

“绝对连续”是经典测度论中一个极为重要的概念。为了研究Ｆｕｚｚｙ测度的扩张问题，王震源在［３］中引进了Ｆｕｚｚｙ测度绝对连续的概念，可以看到它是经典情况的推广。本文在更为广泛的范围内给出了Ｆｕｚｙ测度绝对连续的概念，并得到了一系列等价命题。本文还证明了距离空间中绝对连续性的遗传性。最后还给出了Ｆｕｚｙ测度奇异性的概念，并讨论了Ｆｕｚｙ测度绝对连续性与奇异性的关系     

Sets of Harmonicity for Finely Harmonic Functions

Given an open set U in R ⁿ (n3) and a dense open subset V of U, it is shown that there is a finely harmonic function u on U such that V is the largest open subset of U on which u is harmonic. This result, which establishes the sharpness of a theorem of Fuglede, is obtained following a consideration of fine cluster sets of arbitrary functions.     

多项式Julia集的连续性

本文利用位势理论和复动力系统中的技巧，对多项式Ｊｕｌｉａ集在参数空间的连续性作了完全的刻画．     

Packing Measure Analysis of Harmonic Measure

We show that, for any Jordan domain J in R², harmonic measureis supported by a Borel set of packing dimension 1. We alsoobtain incomplete analogs to the results of Makarov, which connectthe almost everywhere behavior of the derivative near the boundaryfor the conformal mapping function from the unit disk J withthe Hausdorff measure properties of sets supporting the harmonicmeasure.     

Two-Point Projection Estimates for Harmonic Measure

A theorem of Beurling and Nevanlinna gives a lower bound forthe harmonic measure of a compact set in the unit disk in termsof the harmonic measure of a projection of the set. In thispaper, a generalization of that result is proved, involvingthe sum of harmonic measures evaluated at two points. 2000 MathematicsSubject Classification 30C85, 31A15 (primary).     

Lebesgue积分列的等度绝对连续性

给出了Lebesgue积分列的等度绝对连续的一个充分必要条件.     

Absolute Continuity of Quasiconformal Mappings on Curves

We show that a quasiconformal mapping between two proper, locally Ahlfors Q-regular metric spaces, Q > 1, is absolutely continuous on almost every curve. We further relax the limes superior in the definition of quasiconformality to a limes inferior and verify that exceptional sets analogous to the Euclidean setting can be allowed. The authors were supported by grants from the Swiss NSF and the Academy of Finland. Part of the research was done while S.R. was visiting at the University of Bern. Received: August 2005 Accepted: January 2006     

On the Exceptional Set for Absolute Continuity of Bernoulli Convolutions

We prove that the set of exceptional \({\lambda\in (1/2,1)}\) such that the associated Bernoulli convolution is singular has zero Hausdorff dimension, and likewise for biased Bernoulli convolutions, with the exceptional set independent of the bias. This improves previous results by Erdös, Kahane, Solomyak, Peres and Schlag, and Hochman. A theorem of this kind is also obtained for convolutions of homogeneous self-similar measures. The proofs are very short, and rely on old and new results on the dimensions of self-similar measures and their convolutions, and the decay of their Fourier transform.