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Mihalis Mourgoglou - One of the best experts on this subject based on the ideXlab platform.
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Harmonic Measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem
Inventiones Mathematicae, 2020Co-Authors: Jonas Azzam, Mihalis Mourgoglou, Steve Hofmann, Jose Maria Martell, Xavier TolsaAbstract:It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak- $$A_\infty $$ property) of Harmonic Measure with respect to surface Measure, on the boundary of an open set $$ \Omega \subset \mathbb {R}^{n+1}$$ with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in $$\Omega $$ , with data in $$L^p(\partial \Omega )$$ for some $$p<\infty $$ . In this paper, we give a geometric characterization of the weak- $$A_\infty $$ property, of Harmonic Measure, and hence of solvability of the $$L^p$$ Dirichlet problem for some finite p. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors–David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors–David bounds); moreover, the examples show that the upper and lower Ahlfors–David bounds are each quantitatively sharp.
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a two phase free boundary problem for Harmonic Measure and uniform rectifiability
Transactions of the American Mathematical Society, 2020Co-Authors: Jonas Azzam, Mihalis Mourgoglou, Xavier TolsaAbstract:We assume that $\Omega_1, \Omega_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ are two disjoint domains whose complements satisfy the capacity density condition and the intersection of their boundaries $F$ has positive Harmonic Measure. Then we show that in a fixed ball $B$ centered on $F$, if the Harmonic Measure of $\Omega_1$ satisfies a scale invariant $A_\infty$-type condition with respect to the Harmonic Measure of $\Omega_2$ in $B$, then there exists a uniformly $n$-rectifiable set $\Sigma$ so that the Harmonic Measure of $\Sigma \cap F$ contained in $B$ is bounded below by a fixed constant independent of $B$. A remarkable feature of this result is that the Harmonic Measures do not need to satisfy any doubling condition. In the particular case that $\Omega_1$ and $\Omega_2$ are complementary NTA domains, we obtain a geometric characterization of the $A_\infty$ condition between the respective Harmonic Harmonic Measures of $\Omega_1$ and $\Omega_2$.
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Harmonic Measure and riesz transform in uniform and general domains
Crelle's Journal, 2020Co-Authors: Mihalis Mourgoglou, Xavier TolsaAbstract:Let $\Omega\subsetneq\mathbb R^{n+1}$ be open and let $\mu$ be some Measure supported on $\partial\Omega$ such that $\mu(B(x,r))\leq C\,r^n$ for all $x\in\mathbb R^{n+1}$, $r>0$. We show that if the Harmonic Measure in $\Omega$ satisfies some scale invariant $A_\infty$ type conditions with respect to $\mu$, then the $n$-dimensional Riesz transform $$R_\mu f(x) = \int \frac{x-y}{|x-y|^{n+1}}\,f(y)\,d\mu(y)$$ is bounded in $L^2(\mu)$. We do not assume any doubling condition on $\mu$. We also consider the particular case when $\Omega$ is a bounded uniform domain. To this end, we need first to obtain sharp estimates that relate the Harmonic Measure and the Green function in this type of domains, which generalize classical results by Jerison and Kenig for the well-known class of NTA domains.
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absolute continuity of Harmonic Measure for domains with lower regular boundaries
Advances in Mathematics, 2019Co-Authors: Murat Akman, Jonas Azzam, Mihalis MourgoglouAbstract:Abstract We study absolute continuity of Harmonic Measure with respect to surface Measure on domains Ω that have large complements. We show that if Γ ⊂ R d + 1 is Ahlfors d-regular and splits R d + 1 into two NTA domains, then ω Ω ≪ H d on Γ ∩ ∂ Ω . This result is a natural generalization of a result of Wu in [49] . We also prove that almost every point in Γ ∩ ∂ Ω is a cone point if Γ is a Lipschitz graph. Combining these results and a result from [8] , we characterize sets of absolute continuity (with finite H d -Measure if d > 1 ) for domains with large complements both in terms of the cone point condition and in terms of the rectifiable structure of the boundary. Even in the plane, this extends the results of McMillan in [38] and Pommerenke in [43] , which were only known for simply connected planar domains. Finally, we also show our first result holds for elliptic Measure associated with real second order divergence form elliptic operators with a mild assumption on the gradient of the matrix.
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Harmonic Measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem part ii
arXiv: Classical Analysis and ODEs, 2018Co-Authors: Jonas Azzam, Mihalis Mourgoglou, Xavier TolsaAbstract:Let $\Omega\subset\mathbb R^{n+1}$ be an open set with $n$-AD-regular boundary. In this paper we prove that if the Harmonic Measure for $\Omega$ satisfies the so-called weak-$A_\infty$ condition, then $\Omega$ satisfies a suitable connectivity condition, namely the weak local John condition. Together with other previous results by Hofmann and Martell, this implies that the weak-$A_\infty$ condition for Harmonic Measure holds if and only if $\partial\Omega$ is uniformly $n$-rectifiable and the weak local John condition is satisfied. This yields the first geometric characterization of the weak-$A_\infty$ condition for Harmonic Measure, which is important because of its connection with the Dirichlet problem for the Laplace equation.
Jonas Azzam - One of the best experts on this subject based on the ideXlab platform.
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Harmonic Measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem
Inventiones Mathematicae, 2020Co-Authors: Jonas Azzam, Mihalis Mourgoglou, Steve Hofmann, Jose Maria Martell, Xavier TolsaAbstract:It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak- $$A_\infty $$ property) of Harmonic Measure with respect to surface Measure, on the boundary of an open set $$ \Omega \subset \mathbb {R}^{n+1}$$ with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in $$\Omega $$ , with data in $$L^p(\partial \Omega )$$ for some $$p<\infty $$ . In this paper, we give a geometric characterization of the weak- $$A_\infty $$ property, of Harmonic Measure, and hence of solvability of the $$L^p$$ Dirichlet problem for some finite p. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors–David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors–David bounds); moreover, the examples show that the upper and lower Ahlfors–David bounds are each quantitatively sharp.
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a two phase free boundary problem for Harmonic Measure and uniform rectifiability
Transactions of the American Mathematical Society, 2020Co-Authors: Jonas Azzam, Mihalis Mourgoglou, Xavier TolsaAbstract:We assume that $\Omega_1, \Omega_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ are two disjoint domains whose complements satisfy the capacity density condition and the intersection of their boundaries $F$ has positive Harmonic Measure. Then we show that in a fixed ball $B$ centered on $F$, if the Harmonic Measure of $\Omega_1$ satisfies a scale invariant $A_\infty$-type condition with respect to the Harmonic Measure of $\Omega_2$ in $B$, then there exists a uniformly $n$-rectifiable set $\Sigma$ so that the Harmonic Measure of $\Sigma \cap F$ contained in $B$ is bounded below by a fixed constant independent of $B$. A remarkable feature of this result is that the Harmonic Measures do not need to satisfy any doubling condition. In the particular case that $\Omega_1$ and $\Omega_2$ are complementary NTA domains, we obtain a geometric characterization of the $A_\infty$ condition between the respective Harmonic Harmonic Measures of $\Omega_1$ and $\Omega_2$.
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Dimension Drop for Harmonic Measure on Ahlfors Regular Boundaries
Potential Analysis, 2019Co-Authors: Jonas AzzamAbstract:We show that given a domain Ω ⊆ ℝ d + 1 ${\Omega }\subseteq \mathbb {R}^{d+1}$ with uniformly non-flat Ahlfors s -regular boundary with s ≥ d , the dimension of its Harmonic Measure is strictly less than s .
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absolute continuity of Harmonic Measure for domains with lower regular boundaries
Advances in Mathematics, 2019Co-Authors: Murat Akman, Jonas Azzam, Mihalis MourgoglouAbstract:Abstract We study absolute continuity of Harmonic Measure with respect to surface Measure on domains Ω that have large complements. We show that if Γ ⊂ R d + 1 is Ahlfors d-regular and splits R d + 1 into two NTA domains, then ω Ω ≪ H d on Γ ∩ ∂ Ω . This result is a natural generalization of a result of Wu in [49] . We also prove that almost every point in Γ ∩ ∂ Ω is a cone point if Γ is a Lipschitz graph. Combining these results and a result from [8] , we characterize sets of absolute continuity (with finite H d -Measure if d > 1 ) for domains with large complements both in terms of the cone point condition and in terms of the rectifiable structure of the boundary. Even in the plane, this extends the results of McMillan in [38] and Pommerenke in [43] , which were only known for simply connected planar domains. Finally, we also show our first result holds for elliptic Measure associated with real second order divergence form elliptic operators with a mild assumption on the gradient of the matrix.
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dimension drop for Harmonic Measure on ahlfors regular boundaries
arXiv: Classical Analysis and ODEs, 2018Co-Authors: Jonas AzzamAbstract:We show that given a domain $\Omega\subseteq \mathbb{R}^{d+1}$ with uniformly non-flat Ahlfors $s$-regular boundary and $s\geq d$, the dimension of its Harmonic Measure is strictly less than $s$.
Xavier Tolsa - One of the best experts on this subject based on the ideXlab platform.
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Harmonic Measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem
Inventiones Mathematicae, 2020Co-Authors: Jonas Azzam, Mihalis Mourgoglou, Steve Hofmann, Jose Maria Martell, Xavier TolsaAbstract:It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak- $$A_\infty $$ property) of Harmonic Measure with respect to surface Measure, on the boundary of an open set $$ \Omega \subset \mathbb {R}^{n+1}$$ with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in $$\Omega $$ , with data in $$L^p(\partial \Omega )$$ for some $$p<\infty $$ . In this paper, we give a geometric characterization of the weak- $$A_\infty $$ property, of Harmonic Measure, and hence of solvability of the $$L^p$$ Dirichlet problem for some finite p. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors–David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors–David bounds); moreover, the examples show that the upper and lower Ahlfors–David bounds are each quantitatively sharp.
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a two phase free boundary problem for Harmonic Measure and uniform rectifiability
Transactions of the American Mathematical Society, 2020Co-Authors: Jonas Azzam, Mihalis Mourgoglou, Xavier TolsaAbstract:We assume that $\Omega_1, \Omega_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ are two disjoint domains whose complements satisfy the capacity density condition and the intersection of their boundaries $F$ has positive Harmonic Measure. Then we show that in a fixed ball $B$ centered on $F$, if the Harmonic Measure of $\Omega_1$ satisfies a scale invariant $A_\infty$-type condition with respect to the Harmonic Measure of $\Omega_2$ in $B$, then there exists a uniformly $n$-rectifiable set $\Sigma$ so that the Harmonic Measure of $\Sigma \cap F$ contained in $B$ is bounded below by a fixed constant independent of $B$. A remarkable feature of this result is that the Harmonic Measures do not need to satisfy any doubling condition. In the particular case that $\Omega_1$ and $\Omega_2$ are complementary NTA domains, we obtain a geometric characterization of the $A_\infty$ condition between the respective Harmonic Harmonic Measures of $\Omega_1$ and $\Omega_2$.
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Harmonic Measure and riesz transform in uniform and general domains
Crelle's Journal, 2020Co-Authors: Mihalis Mourgoglou, Xavier TolsaAbstract:Let $\Omega\subsetneq\mathbb R^{n+1}$ be open and let $\mu$ be some Measure supported on $\partial\Omega$ such that $\mu(B(x,r))\leq C\,r^n$ for all $x\in\mathbb R^{n+1}$, $r>0$. We show that if the Harmonic Measure in $\Omega$ satisfies some scale invariant $A_\infty$ type conditions with respect to $\mu$, then the $n$-dimensional Riesz transform $$R_\mu f(x) = \int \frac{x-y}{|x-y|^{n+1}}\,f(y)\,d\mu(y)$$ is bounded in $L^2(\mu)$. We do not assume any doubling condition on $\mu$. We also consider the particular case when $\Omega$ is a bounded uniform domain. To this end, we need first to obtain sharp estimates that relate the Harmonic Measure and the Green function in this type of domains, which generalize classical results by Jerison and Kenig for the well-known class of NTA domains.
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the mutual singularity of Harmonic Measure and hausdorff Measure of codimension smaller than one
arXiv: Classical Analysis and ODEs, 2019Co-Authors: Xavier TolsaAbstract:Let $\Omega\subset\mathbb R^{n+1}$ be open and let $E\subset \partial\Omega$ with $0
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on tsirelson s theorem about triple points for Harmonic Measure
International Mathematics Research Notices, 2018Co-Authors: Xavier Tolsa, Alexander VolbergAbstract:A theorem of Tsirelson from 1997 asserts that given three disjoint domains in $\mathbb R^{n+1}$, the set of triple points belonging to the intersection of the three boundaries where the three corresponding Harmonic Measures are mutually absolutely continuous has null Harmonic Measure. The original proof by Tsirelson is based on the fine analysis of filtrations for Brownian and Walsh-Brownian motions and can not be translated into potential theory arguments. In the present paper we give a purely analytical proof of the same result.
Steve Hofmann - One of the best experts on this subject based on the ideXlab platform.
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Harmonic Measure and quantitative connectivity geometric characterization of the l p solvability of the dirichlet problem
Inventiones Mathematicae, 2020Co-Authors: Jonas Azzam, Mihalis Mourgoglou, Steve Hofmann, Jose Maria Martell, Xavier TolsaAbstract:It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak- $$A_\infty $$ property) of Harmonic Measure with respect to surface Measure, on the boundary of an open set $$ \Omega \subset \mathbb {R}^{n+1}$$ with Ahlfors–David regular boundary, is equivalent to the solvability of the Dirichlet problem in $$\Omega $$ , with data in $$L^p(\partial \Omega )$$ for some $$p<\infty $$ . In this paper, we give a geometric characterization of the weak- $$A_\infty $$ property, of Harmonic Measure, and hence of solvability of the $$L^p$$ Dirichlet problem for some finite p. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors–David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors–David bounds); moreover, the examples show that the upper and lower Ahlfors–David bounds are each quantitatively sharp.
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quantitative absolute continuity of Harmonic Measure and the dirichlet problem a survey of recent progress
Acta Mathematica Sinica, 2019Co-Authors: Steve HofmannAbstract: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 bound¬ary 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 Lp(∂ Ω) for some p < ∞. Drawing an analogy to the famous Wiener criterion, which characterizes the domains in which the classical Dirichlet problem, with contin¬uous boundary data, can be solved, one may seek to characterize the open sets for which Lp 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 charac¬terization of the weak-A∞ property, and hence of solvability of the Lp 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.
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quantitative absolute continuity of Harmonic Measure and the dirichlet problem a survey of recent progress
Acta Mathematica Sinica, 2019Co-Authors: Steve HofmannAbstract: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 bound¬ary 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 Lp(∂ Ω) for some p < ∞. Drawing an analogy to the famous Wiener criterion, which characterizes the domains in which the classical Dirichlet problem, with contin¬uous boundary data, can be solved, one may seek to characterize the open sets for which Lp solvability holds, thus allowing for singular boundary data.
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bmo solvability and absolute continuity of Harmonic Measure
Journal of Geometric Analysis, 2018Co-Authors: Steve HofmannAbstract: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.
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a sufficient geometric criterion for quantitative absolute continuity of Harmonic Measure
2017Co-Authors: Steve Hofmann, Jose Maria MartellAbstract:Let $\Omega\subset \mathbb{R}^{n+1}$, $n\ge 2$, be an open set, not necessarily connected, with an $n$-dimensional uniformly rectifiable boundary. We show that Harmonic Measure for $\Omega$ is weak-$A_\infty$ with respect to surface Measure on $\partial\Omega$, provided that $\Omega$ satisfies a certain weak version of a local John condition.
Serguei Popov - One of the best experts on this subject based on the ideXlab platform.
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conditioned two dimensional simple random walk green s function and Harmonic Measure
Journal of Theoretical Probability, 2021Co-Authors: Serguei PopovAbstract:We study the Doob’s h-transform of the two-dimensional simple random walk with respect to its potential kernel, which can be thought of as the two-dimensional simple random walk conditioned on never hitting the origin. We derive an explicit formula for the Green’s function of this random walk and also prove a quantitative result on the speed of convergence of the (conditional) entrance Measure to the Harmonic Measure (for the conditioned walk) on a finite set.
