Jonas Azzam

• PhD Mathematics
• Lecturer at University of Edinburgh

We prove Lp 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 Lp norm of the gradient of a Sobolev function f : E → R is comparable to the Lp norm of a new square function measuring both the affine...

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 \(0<t<r\) satisfying \(\mathbf{H}^{d}_{\infty}(E\cap B(y,t))<(2t)^{d}-\epsilon(2r)^d\) is a Carleson set for every \(\epsilon>0\). To prove this, we generalize...

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...

We show that given a domain \({\Omega }\subseteq \mathbb {R}^{d+1}\) with uniformly non-flat Ahlfors s-regular boundary with s ≥ d, the dimension of its harmonic measure is strictly less than s.

We give a characterization of $L^{p}(\sigma)$ for uniformly rectifiable measures $\sigma$ using Tolsa's $\alpha$-numbers, by showing, for $1<p<\infty$ and $f\in L^{p}(\sigma)$, that \[ \lVert f\rVert_{L^{p}(\sigma)}\sim \left\lVert\left(\int_{0}^{\infty} \left(\alpha_{f\sigma}(x,r)+|f|_{x,r}\alpha_{\sigma}(x,r)\right)^2\ \frac{dr}{r} \right)^{\frac...

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 $0<t<r$ satisfying $\mathscr{H}^{d}_{\infty}(E\cap B(y,t))<(2t)^{d}-\varepsilon(2r)^d$ is a Carleson set for every $\varepsilon>0$. To prove this, we generalize a result...

We characterize Radon measures μ \mu in R n \mathbb {R}^{n} that are d d -rectifiable in the sense that their supports are covered up to μ \mu -measure zero by countably many d d -dimensional Lipschitz images and μ ≪ H d \mu \ll \mathcal {H}^{d} . The characterization is in terms of a Jones function involving the so-called α \alpha -numbers. This a...

We show that any $d$-Ahlfors regular subset of $\mathbb{R}^{n}$ supporting a weak $(1,d)$-Poincar\'e inequality with respect to surface measure is uniformly rectifiable.

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...

We study how generalized Jones $\beta$-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 Corona decompositions for harmonic measure if and only if the square sum $\beta_{\partial\Omega}$ of the genera...

We generalize some characterizations of uniformly rectifiable (UR) sets to sets whose Hausdorff content is lower regular (and in particular, do not need to be Ahlfors regular). For example, David and Semmes showed that, given an Ahlfors $d$-regular set $E$, if we consider the set $\mathscr{B}$ of surface cubes (in the sense of Christ and David) nea...

We show that given a domain $\Omega\subseteq \mathbb{R}^{d+1}$ with uniformly non-flat Ahlfors $s$-regular boundary with $s\geq d$, the dimension of its harmonic measure is strictly less than $s$.

We characterize Radon measures $\mu$ in $\mathbb{R}^{n}$ that are $d$-rectifiable in the sense that their supports are covered up to $\mu$-measure zero by countably many $d$-dimensional Lipschitz graphs and $\mu \ll \mathcal{H}^{d}$. The characterization is in terms of a Jones function involving the so-called $\alpha$-numbers. This answers a questi...

We show that if $0<t<s\leq n-1$, $\Omega\subseteq \mathbb{R}^{n}$ with lower $s$-content regular complement, and $z\in \Omega$, there is a chord-arc domain $\Omega_{z}\subseteq \Omega $ with center $z$ so that $\mathscr{H}^{t}_{\infty}(\partial\Omega_{z}\cap \partial\Omega)\gtrsim_{t} \textrm{dist}(z,\Omega^{c})^{t}$. This was originally shown by K...

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $\beta$-numbers. These $\beta$-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones' result is a quantitative way of saying that a curve is rectifiable...

Let Ω ⊂ 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∞ condition, then Ω 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∞ condition f...

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 Marte...

We show that for uniform domains Ω ⊆ ℝd+1 whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to α-dimensional Hausdorff measure unless α ≤ d. We employ a lemma that shows that, at almost every non-degenerate point, we may find a tangent measure of harmonic measure w...

For domains with Ahlfors-David regular boundaries, it is known that the $A_{\infty}$ property of harmonic measure implies uniform rectifiability of the boundary \cite{MT15,HLMN17}. Since $A_{\infty}$-weights are doubling, this also implies the domain is semi-uniform if the domain is John by the work of Aikawa and Hirata \cite{AH08}. In this paper,...

We assume that $\Omega_1, \Omega_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ are two disjoint domains whose complements satisfy a uniform thickness 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 $\om_1$ satisfies a scale invariant $A...

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 invarian...

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 measures of elliptic measures in arbitrary domains associated with (possibly non-symmetric) elliptic operators...

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 measures of elliptic measures in arbitrary domains associated with (possibly non-symmetric) elliptic operators...

We study the structure of the support of a doubling measure by analyzing its self-similarity properties, which we estimate using a variant of the (Formula presented.) Wasserstein distance. We show that a measure satisfying certain self-similarity conditions admits a unique (up to multiplication by a constant) flat tangent measure at almost every po...

Let $\Omega\subset\mathbb R^{n+1}$, $n\geq2$, 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, merely bounded and possibly non-symmetric coefficients, which are also locally Lipschitz and satisfy suitabl...

Let $\Omega\subset\mathbb{R}^{n+1}$, $n\geq2$, 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, merely bounded and possibly non-symmetric coefficients, which are also locally Lipschitz and satisfy suitab...

We show that if Ω is an NTA domain with harmonic measure ω and E⊆∂Ω is contained in an Ahlfors regular set, then \(\omega |_{E}\ll \mathcal {H}^{d}|_{E}\). Moreover, this holds quantitatively in the sense that for all τ>0ω obeys an A ∞ -type condition with respect to \(\mathcal {H}^{d}|_{E^{\prime }}\), where E ′⊆E is so that ω(E∖E ′)<τ ω(E), even...

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 corresponding Lipschitz constant cannot be bounded.

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 corresponding Lipschitz constant cannot be bounded.

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 improves on our previous result which assumed that the boundaries satisfied the capacity density condition.

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 improves on our previous result which assumed that the boundaries satisfied the capacity density condition.

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $\beta$-numbers. These $\beta$-numbers are geometric quantities measuring how far a given set deviates from a best fitting line at each scale and location. Jones' result is a quantitative way of saying that a curve is rectifiable...

A theorem of David and Jerison asserts that harmonic measure is absolutely continuous with respect to surface measure in NTA domains with Ahlfors regular boundaries. We prove that this fails in high dimensions if we relax the Ahlfors regularity assumption by showing that, for each $d>1$, there exists a Reifenberg flat domain $\Omega\subset \mathbb{...

A natural quantity that measures how well a map f\colon \mathbb{R}^{d}\rightarrow \mathbb{R}^{D} is approximated by an affine transformation is \omega_{f}(x,r)=\inf_{A}\Big(\, \mathrlap{\,-}{\int}_{B(x,r)}\Big(\frac{|f-A|}{|A'|r}\Big)^{2}\,\Big)^{{1}/{2}}, where the infimum ranges over all non-zero affine transformations A . This is natural insofar...

In the present paper we prove that for any open connected set $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial \Omega$ with $0<\mathcal{H}^n(E)<\infty$ absolute continuity of the harmonic measure $\omega$ with respect to the Hausdorff measure on $E$ implies that $\omega|_E$ is rectifiable.

We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $ \mathbb{R}^{d+1}$ into two NTA domains then $\omega_{\Omega}\ll \mathscr{H}^{d}$ on $\Gamma\cap \partial\Omega$. This result is a natur...

We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $ \mathbb{R}^{d+1}$ into two NTA domains then $\omega_{\Omega}\ll \mathscr{H}^{d}$ on $\Gamma\cap \partial\Omega$. This result is a natur...

We show that if $\Omega\subset\mathbb R^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $\textrm{VMO}(\sigma)$, where $\sigma$ is the surface measure of $\Omega$, then the outer unit normal to $\partial\Omega$ belongs to $\textrm{VMO}(\sigma)$ too. The analogous result fails for dimensi...

We show that if $\Omega\subset\mathbb R^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $\textrm{VMO}(\sigma)$, where $\sigma$ is the surface measure of $\Omega$, then the outer unit normal to $\partial\Omega$ belongs to $\textrm{VMO}(\sigma)$ too. The analogous result fails for dimensi...

Garnett, Killip, and Schul have exhibited a doubling measure $\mu$ with support equal to $\mathbb{R}^{d}$ which is $1$-rectifiable, meaning there are countably many curves $\Gamma_{i}$ of finite length for which $\mu(\mathbb{R}^{d}\backslash \bigcup \Gamma_{i})=0$. In this note, we characterize when a doubling measure $\mu$ with support equal to a...

We show that, for disjoint domains in the Euclidean space whose boundaries satisfy a non-degeneracy condition, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries.

We show that, for disjoint domains in the Euclidean space whose boundaries satisfy a non-degeneracy condition, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries.

In the present paper we sketch the proof of the fact that for any open connected set $\Omega\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial \Omega$ with $0<\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic measure $\omega$ with respect to the Hausdorff measure on $E$ implies that $\omega|_E$ is rectifiable.

For uniform domains $\Omega\subseteq \mathbb{R}^{d+1}$ satisfying the capacity density condition (or the CDC), we show that harmonic measure has a tangent measure whose support is the boundary of a uniform domain with the CDC. More importantly, the harmonic measure on these domains can be expressed as the weak limit of blown-up copies of the origin...

We show that if $n\geq 2$, $\Omega\subset \mathbb R^{n+1}$ is a connected domain with porous boundary, and $E\subset \partial\Omega$ is a set of finite and positive Hausdorff $H^{n}$-measure upon which the harmonic measure $\omega$ is absolutely continuous with respect to $H^{n}$, then $\omega|_E$ is concentrated on an $n$-rectifiable set.

In a recent paper, Cs\"ornyei and Wilson prove that curves in Eucilidean space of $\sigma$-finite length have tangents on a set of positive measure. They also show that a higher dimensional analogue of this result is not possible without some additional assumptions. In this note, we show that if the boundary of a corkscrew domain in $\mathbb{R}^{d+...

In a recent paper, Cs\"ornyei and Wilson prove that curves in Euclidean space of $\sigma$-finite length have tangents on a set of positive $\mathscr{H}^{1}$-measure. They also show that a higher dimensional analogue of this result is not possible without some additional assumptions. In this note, we show that if $\Sigma\subseteq \mathbb{R}^{d+1}$ h...

We show that a Radon measure $\mu$ in $\mathbb R^d$ which is absolutely continuous with respect to the $n$-dimensional Hausdorff measure $H^n$ is $n$-rectifiable if the so called Jones' square function is finite $\mu$-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give...

Let $\mu$ be a doubling measure in $\mathbb{R}^n$. We investigate quantitative relations between the rectifiability of $\mu$ and its distance to flat measures. More precisely, for $x$ in the support $\Sigma$ of $\mu$ and $r > 0$, we introduce a number $\alpha(x,r)\in (0,1]$ that measures, in terms of a variant of the $L^1$-Wasserstein distance, the...

We show that if $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 1$, is a uniform domain (aka 1-sided NTA domain), i.e., a domain which enjoys interior Corkscrew and Harnack Chain conditions, then uniform rectifiability of the boundary of $\Omega$ implies the existence of exterior Corkscrew points at all scales, so that in fact, $\Omega$ is a chord-arc do...

A quasiplane $f(V)$ is the image of an $n$-dimensional Euclidean subspace $V$ of ${\Bbb R}^N$ ($1\leq n\leq N-1$) under a quasiconformal map $f:{\Bbb R}^N\to{\Bbb R}^N$ . We give sufficient conditions in terms of the weak quasisymmetry constant of the underlying map for a quasiplane to be a bi-Lipschitz $n$-manifold and for a quasiplane to have big...

A natural quantity that measures how well a map $f:\mathbb{R}^{d}\rightarrow \mathbb{R}^{D}$ is approximated by an affine transformation is \[\omega_{f}(x,r)=\inf_{A}\left(\frac{1}{|B(x,r)|}\int_{B(x,r)}\left(\frac{|f-A|}{|A'|r}\right)^{2}\right)^{\frac{1}{2}},\] where the infimum ranges over all non constant affine transformations. This is natural...

For a compact connected set $X\subseteq \ell^{\infty}$, we define a quantity $\beta'(x,r)$ that measures how close $X$ may be approximated in a ball $B(x,r)$ by a geodesic curve. We then show there is $c>0$ so that if $\beta'(x,r)>\beta>0$ for all $x\in X$ and $r<r_{0}$, then $\dim X>1+c\beta^{2}$. This generalizes a theorem of Bishop and Jones and...

The purpose of this note is to point out a simple consequence of some earlier work of the authors, "Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps". For $f$, a Lipschitz function from a Euclidean space into a metric space, we give quantitative estimates for how often the pullback of the metric under $f$ is appro...

We consider a number of uniqueness questions for several wide classes of active scalar equations, unifying and generalizing the techniques of several authors. As special cases of our results, we provide a significantly simplified proof to the known uniqueness result for the 2D Euler equations in $L^1 \cap BMO$ and provide a mild improvement to the...

We prove a global implicit function theorem. In particular we show that any Lipschitz map \({f : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n}}\) (with n-dim. image) can be precomposed with a bi-Lipschitz map \({\bar{g} : \mathbb{R}^{n} \times \mathbb{R}^{m} \rightarrow \mathbb{R}^{n} \times \mathbb{R}^{m}}\) such that \({f \circ...

The purpose of this note is to point out a simple consequence of some earlier work of the authors, "Hard Sard: Quantitative implicit function and extension theorems for Lipschitz maps". For $f$, a Lipschitz function from a Euclidean space into a metric space, we give quantitative estimates for how often the pullback of the metric under $f$ is appro...

For a given connected set $\Gamma$ in $d-$dimensional Euclidean space, we construct a connected set $\tilde\Gamma\supset \Gamma$ such that the two sets have comparable Hausdorff length, and the set $\tilde\Gamma$ has the property that it is quasiconvex, i.e. any two points $x$ and $y$ in $\tilde\Gamma$ can be connected via a path, all of which is i...

We prove that there exists $M>0$ such that for any closed rectifiable curve $\Gamma$ in Hilbert space, almost every point in $\Gamma$ is contained in a countable union of $M$ chord-arc curves whose total length is no more than $M$ times the length of $\Gamma$.

On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy Eϕ and conformal Laplacian ∆ϕ for a given conformal factor ϕ, based on the corresponding notions in Riemannian geometry in dimension n = 2. We derive a differential equation that describes the dependence of the effective resistances of Eϕ on ϕ. We show that t...