Jonas Azzam
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View article: Quantitative differentiability on uniformly rectifiable sets
Quantitative differentiability on uniformly rectifiable sets Open
We prove $L^p$ quantitative differentiability estimates for functions defined on uniformly rectifiable subsets of the Euclidean space. More precisely, we show that a Dorronsoro-type theorem holds in this context: the $L^p$ norm of the grad…
View article: The weak lower density condition and uniform rectifiability
The weak lower density condition and uniform rectifiability Open
We show that an Ahlfors \(d\)-regular set \(E\) in \(\mathbb{R}^{n}\) is uniformly rectifiable if the set of pairs \((x,r)\in E\times (0,\infty)\) for which there exists \(y \in B(x,r)\) and \(00\). To prove this, we generalize a result of…
View article: Uniform Rectifiability, Elliptic Measure, Square Functions, and <i>ε</i>-Approximability Via an ACF Monotonicity Formula
Uniform Rectifiability, Elliptic Measure, Square Functions, and <i>ε</i>-Approximability Via an ACF Monotonicity Formula Open
Let $\Omega \subset{{\mathbb{R}}}^{n+1}$, $n\geq 2$, be an open set with Ahlfors regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with re…
View article: Poincaré inequalities and uniform rectifiability
Poincaré inequalities and uniform rectifiability Open
We show that any d -Ahlfors regular subset of \mathbb R^n supporting a weak (1, d) -Poincaré inequality with respect to surface measure is uniformly rectifiable.
View article: An $α$-number characterization of $L^{p}$ spaces on uniformly rectifiable sets
An $α$-number characterization of $L^{p}$ spaces on uniformly rectifiable sets Open
We give a characterization of $L^{p}(σ)$ for uniformly rectifiable measures $σ$ using Tolsa's $α$-numbers, by showing, for $1
View article: The weak lower density condition and uniform rectifiability
The weak lower density condition and uniform rectifiability Open
We show that an Ahlfors $d$-regular set $E$ in $\mathbb{R}^{n}$ is uniformly rectifiable if the set of pairs $(x,r)\in E\times (0,\infty)$ for which there exists $y \in B(x,r)$ and $00$. To prove this, we generalize a result of Schul by pr…
View article: Accessible parts of the boundary for domains with lower content regular complements
Accessible parts of the boundary for domains with lower content regular complements Open
We show that if $0n-p+β$ was necessary. Thus, the combination of these results gives a characterization of which domains support pointwise $(p,β)$-Hardy inequalities for $β
View article: Harmonic Measure and the Analyst's Traveling Salesman Theorem
Harmonic Measure and the Analyst's Traveling Salesman Theorem Open
We study how generalized Jones $β$-numbers relate to harmonic measure. Firstly, we generalize a result of Garnett, Mourgoglou and Tolsa by showing that domains in $\mathbb{R}^{d+1}$ whose boundaries are lower $d$-content regular admit Coro…
View article: On a two-phase problem for harmonic measure in general domains
On a two-phase problem for harmonic measure in general domains Open
We show that, for disjoint domains in the Euclidean space, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries. This i…
View article: Characterization of rectifiable measures in terms of $α$-numbers
Characterization of rectifiable measures in terms of $α$-numbers Open
We characterize Radon measures $μ$ in $\mathbb{R}^{n}$ that are $d$-rectifiable in the sense that their supports are covered up to $μ$-measure zero by countably many $d$-dimensional Lipschitz graphs and $μ\ll \mathcal{H}^{d}$. The characte…
View article: Characterization of rectifiable measures in terms of $\alpha$-numbers
Characterization of rectifiable measures in terms of $\alpha$-numbers Open
We characterize Radon measures $\\mu$ in $\\mathbb{R}^{n}$ that are\n$d$-rectifiable in the sense that their supports are covered up to\n$\\mu$-measure zero by countably many $d$-dimensional Lipschitz graphs and $\\mu\n\\ll \\mathcal{H}^{d…
View article: Some remarks on the Lipschitz regularity of Radon transforms
Some remarks on the Lipschitz regularity of Radon transforms Open
A set in the Euclidean plane is constructed whose image under the classical Radon transform is Lipschitz in every direction. It is also shown that, under mild hypotheses, for any such set the function which maps a direction to the correspo…
View article: Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$ solvability of the Dirichlet problem. Part II
Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$ solvability of the Dirichlet problem. Part II Open
Let $Ω\subset\mathbb R^{n+1}$ be an open set with $n$-AD-regular boundary. In this paper we prove that if the harmonic measure for $Ω$ satisfies the so-called weak-$A_\infty$ condition, then $Ω$ satisfies a suitable connectivity condition,…
View article: Tangent measures and absolute continuity of harmonic measure
Tangent measures and absolute continuity of harmonic measure Open
We show that for uniform domains \Omega\subseteq \mathbb R^{d+1} whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to \alpha -dimensional Hausdorff measur…
View article: Semi-uniform domains and the $A_{\infty}$ property for harmonic measure
Semi-uniform domains and the $A_{\infty}$ property for harmonic measure Open
We study the properties of harmonic measure in semi-uniform domains. Aikawa and Hirata showed in \cite{AH08} that, for John domains satisfying the capacity density condition (CDC), the doubling property for harmonic measure is equivalent t…
View article: A two-phase free boundary problem for harmonic measure and uniform\n rectifiability
A two-phase free boundary problem for harmonic measure and uniform\n rectifiability Open
We assume that $\\Omega_1, \\Omega_2 \\subset \\mathbb{R}^{n+1}$, $n \\geq 1$ are\ntwo disjoint domains whose complements satisfy the capacity density condition\nand the intersection of their boundaries $F$ has positive harmonic measure.\n…
View article: A two-phase free boundary problem for harmonic measure and uniform\n rectifiability
A two-phase free boundary problem for harmonic measure and uniform\n rectifiability Open
We assume that $\\Omega_1, \\Omega_2 \\subset \\mathbb{R}^{n+1}$, $n \\geq 1$ are\ntwo disjoint domains whose complements satisfy the capacity density condition\nand the intersection of their boundaries $F$ has positive harmonic measure.\n…
View article: Tangent measures of elliptic harmonic measure and applications
Tangent measures of elliptic harmonic measure and applications Open
Tangent measure and blow-up methods, are powerful tools for understanding the relationship between the infinitesimal structure of the boundary of a domain and the behavior of its harmonic measure. We introduce a method for studying tangent…
View article: The one-phase problem for harmonic measure in two-sided NTA domains
The one-phase problem for harmonic measure in two-sided NTA domains Open
We show that if $\\Omega\\subset\\mathbb R^3$ is a two-sided NTA domain with\nAD-regular boundary such that the logarithm of the Poisson kernel belongs to\n$\\textrm{VMO}(\\sigma)$, where $\\sigma$ is the surface measure of $\\Omega$, then…
View article: A new characterization of chord-arc domains
A new characterization of chord-arc domains Open
We show that if \Omega \subset \mathbb{R}^{n+1} , n\geq 1 , is a uniform domain (also known as a 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary …
View article: Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability
Mutual Absolute Continuity of Interior and Exterior Harmonic Measure Implies Rectifiability Open
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 rectifiabil…
View article: Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximability via an ACF monotonicity formula
Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximability via an ACF monotonicity formula Open
Let $Ω\subset\mathbb{R}^{n+1}$, $n\geq2$, be an open set with Ahlfors-David regular boundary that satisfies the corkscrew condition. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with real, m…
View article: Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximability
Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximability Open
Let $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 2$, be an open set with Ahlfors-David regular boundary. We consider a uniformly elliptic operator $L$ in divergence form associated with a matrix $A$ with real and merely bounded coefficients whi…
View article: Some remarks on the Lipschitz regularity of Radon transforms
Some remarks on the Lipschitz regularity of Radon transforms Open
A set in the Euclidean plane is constructed whose image under the classical Radon transform is Lipschitz in every direction. It is also shown that, under mild hypotheses, for any such set the function which maps a direction to the correspo…
View article: An Analyst's Traveling Salesman Theorem for sets of dimension larger\n than one
An Analyst's Traveling Salesman Theorem for sets of dimension larger\n than one Open
In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable\ncurves in the plane via a multiscale sum of $\\beta$-numbers. These\n$\\beta$-numbers are geometric quantities measuring how far a given set deviates\nfrom a bes…