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Jonas Azzam

Publications and source records attributed to Jonas Azzam.

At least 19 recordsLinked to original sources
arXiv2025

Smooth extensions of Sobolev boundary data in corkscrew domains with uniformly rectifiable boundaries

Given a corkscrew domain with uniformly rectifiable boundary, we construct a surjective trace map onto the $L^p$ Hajlasz-Sobolev space on the boundary from the space of functions on the domain with $L^p$ norm involving the non-tangential maximal function of the gradient and the conical square function of the Hessian. This fundametally uses the Dorronsoro theorem for UR sets proven in a companion paper.

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arXiv2023

Quantitative differentiability on uniformly rectifiable sets

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 gradient of a Sobolev function $f: E \to \mathbb{R}$ is comparable to the $L^p$ norm of a new square function measuring both the affine deviation of $f$ and how flat the subset $E$ is. A corollary dealing with extensions and traces of Sobolev functions may be found in a companion article.

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arXiv2021

The weak lower density condition and uniform rectifiability

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 $0 0$. To prove this, we generalize a result of Schul by proving, if $X$ is a $C$-doubling metric space, $\varepsilon,ρ\in (0,1)$, $A>1$, and $X_{n}$ is a sequence of maximal $2^{-n}$-separated sets in $X$, and $\mathscr{B}=\{B(x,2^{-n}):x\in X_{n},n\in \mathbb{N}\}$, then \[ \sum \left\{r_{B}^{s}: B\in \mathscr{B}, \frac{\mathscr{H}^{s}_{ρr_{B}}(X\cap AB)}{(2r_{B})^{s}}>1+\varepsilon\right\} \lesssim_{C,A,\varepsilon,ρ,s} \mathscr{H}^{s}(X). \] This is a quantitative version of the classical result that for a metric space $X$ of finite $s$-dimensional Hausdorff measure, the upper $s$-dimensional densities are at most $1$ $\mathscr{H}^{s}$-almost everywhere.

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arXiv2020

An $α$-number characterization of $L^{p}$ spaces on uniformly rectifiable sets

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

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arXiv2020

Quantitative Comparisons of Multiscale Geometric Properties

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) near which $E$ does not look approximately like a union of planes, then $E$ is UR if and only if $\mathscr{B}$ satisfies a Carleson packing condition, that is, for any surface cube $R$, \[ \sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d} \lesssim ({\rm diam} R)^{d}.\] We show that, for lower content regular sets that aren't necessarily Ahlfors regular, if $β_{E}(R)$ denotes the square sum of $β$-numbers over subcubes of $R$ as in the Traveling Salesman Theorem for higher dimensional sets [AS18], then \[ \mathscr{H}^{d}(R)+\sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d}\sim β_{E}(R). \] We prove similar results for other uniform rectifiability critera, such as the Local Symmetry, Local Convexity, and Generalized Weak Exterior Convexity conditions. En route, we show how to construct a corona decomposition of any lower content regular set by Ahlfors regular sets, similar to the classical corona decomposition of UR sets by Lipschitz graphs developed by David and Semmes.

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arXiv2020

Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem

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 $ Ω\subset \mathbb{R}^{n+1}$ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in $Ω$, with data in $L^p(\partialΩ)$ 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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arXiv2020

A two-phase free boundary problem for harmonic measure and uniform rectifiability

We assume that $Ω_1, Ω_2 \subset \mathbb{R}^{n+1}$, $n \geq 1$ 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 $Ω_1$ satisfies a scale invariant $A_\infty$-type condition with respect to the harmonic measure of $Ω_2$ in $B$, then there exists a uniformly $n$-rectifiable set $Σ$ so that the harmonic measure of $Σ\cap F$ 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 $Ω_1$ and $Ω_2$ are complementary NTA domains, we obtain a geometric characterization of the $A_\infty$ condition between the respective harmonic harmonic measures of $Ω_1$ and $Ω_2$.

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arXiv2019

An Analyst's Traveling Salesman Theorem for sets of dimension larger than one

In his 1990 Inventiones paper, P. Jones characterized subsets of rectifiable curves in the plane via a multiscale sum of $β$-numbers. These $β$-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 if and only if it has a tangent at almost every point. Moreover, computing this square sum for a curve returns the length of the curve up to multiplicative constant. K. Okikiolu extended his result from subsets of the plane to subsets of Euclidean space. G. David and S. Semmes extended the discussion to include sets of (integer) dimension larger than one, under the assumption of Ahlfors regularity and using a variant of Jones' $β$ numbers. In this paper we give a version of P. Jones' theorem for sets of arbitrary (integer) dimension lying in Euclidean space. We estimate the $d$-dimensional Hausdorff measure of a set in terms of an analogous sum of $β$-type numbers. There is no assumption of Ahlfors regularity, but rather, only of a lower bound on the Hausdorff content. We adapt David and Semmes' version of Jones' $β$-numbers by redefining them using a Choquet integral. A key tool in the proof is G. David and T. Toro's parametrization of Reifenberg flat sets (with holes).

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arXiv2019

Harmonic Measure and the Analyst's Traveling Salesman Theorem

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 Corona decompositions for harmonic measure if and only if the square sum $β_{\partialΩ}$ of the generalized Jones $β$-numbers is finite. Secondly, for semi-uniform domains with Ahlfors regular boundaries, it is known that uniform rectifiability implies harmonic measure is $A_{\infty}$ for semi-uniform domains, but now we give more explicit dependencies on the $A_{\infty}$-constant in terms of the uniform rectifiability constant. This follows from a more general estimate that does not assume the boundary to be uniformly rectifiable. For general semi-uniform domains, we also show how to bound the harmonic measure of a subset in terms of that sets Hausdorff measure and the square sum of $β$-numbers on that set. Using this, we give estimates on the fluctuation of Green's function in a uniform domain in terms of the $β$-numbers. As a corollary, for bounded NTA domains , if $B_Ω=B(x_Ω,c\mathrm{diam} Ω)$ is so that $2B_Ω\subseteq Ω$, we obtain that \[ (\mathrm{diam} \partialΩ)^{d} + \int_{Ω\backslash B_Ω} \ |\frac{\nabla^2 G_Ω(x_Ω,x)}{G_Ω(x_Ω,x)}\ |^{2} \mathrm{dist}(x,Ω^c)^{3} dx \sim \mathscr{H}^{d}(\partialΩ). \] Secondly, we also use $β$-numbers to estimate how much harmonic measure fails to be $A_{\infty}$-weight for semi-uniform domains with Ahlfors regular boundaries.

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arXiv2018

Semi-uniform domains and the $A_{\infty}$ property for harmonic measure

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 to the domain being semi-uniform. Our first result removes the John condition by showing that any domain satisfying the CDC whose harmonic measure is doubling is semi-uniform. Next, we develop a substitute for some classical estimates on harmonic measure in nontangentially accessible domains that works in semi-uniform domains. We also show that semi-uniform domains with uniformly rectifiable boundary have big pieces of chord-arc subdomains. We cannot hope for big pieces of Lipschitz subdomains (as was shown for chord-arc domains by David and Jerison \cite{DJ90}) due to an example of Hrycak, which we review in the appendix. Finally, we combine these tools to study the $A_{\infty}$-property of harmonic measure. For a domain with Ahlfors-David regular boundary, it was shown by Hofmann and Martell that the $A_{\infty}$ property of harmonic measure implies uniform rectifiability of the boundary \cite{HM15,HLMN17} . Since $A_{\infty}$-weights are doubling, this also implies the domain is semi-uniform. Our final result shows that these two properties, semi-uniformity and uniformly rectifiable boundary, also imply the $A_{\infty}$ property for harmonic measure, thus classifying geometrically all domains for which this holds.

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arXiv2018

Characterization of rectifiable measures in terms of $α$-numbers

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 characterization is in terms of a Jones function involving the so-called $α$-numbers. This answers a question left open in a former work by Azzam, David, and Toro.

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arXiv2018

Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$ solvability of the Dirichlet problem. Part II

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, 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 $\partialΩ$ 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.

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arXiv2017

Tangent measures of elliptic harmonic measure and applications

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 in divergence form whose coefficients have vanishing mean oscillation at the boundary. In this setting, we show the following for domains $ Ω\subset \mathbb{R}^{n+1}$: 1. We extend the results of Kenig, Preiss, and Toro [KPT09] by showing mutual absolute continuity of interior and exterior elliptic measures for {\it any} domains implies the tangent measures are a.e. flat and the elliptic measures have dimension $n$. 2. We generalize the work of Kenig and Toro [KT06] and show that VMO equivalence of doubling interior and exterior elliptic measures for general domains implies the tangent measures are always elliptic polynomials. 3. In a uniform domain that satisfies the capacity density condition and whose boundary is locally finite and has a.e. positive lower $n$-Hausdorff density, we show that if the elliptic measure is absolutely continuous with respect to $n$-Hausdorff measure then the boundary is rectifiable. This generalizes the work of Akman, Badger, Hofmann, and Martell [ABHM17]. Finally, we generalize one of the main results of [Bad11] by showing that if $ω$ is a Radon measure for which all tangent measures at a point are harmonic polynomials vanishing at the origin, then they are all homogeneous harmonic polynomials.

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arXiv2017

Uniform rectifiability, elliptic measure, square functions, and $\varepsilon$-approximability via an ACF monotonicity formula

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, merely bounded and possibly non-symmetric coefficients, which are also locally Lipschitz and satisfy suitable Carleson type estimates. In this paper we show that if $L^*$ is the operator in divergence form associated with the transpose matrix of $A$, then $\partialΩ$ is uniformly $n$-rectifiable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^*v=0$ in $Ω$ is $\varepsilon$-approximmable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^*v=0$ in $Ω$ satisfies a suitable square-function Carleson measure estimate. Moreover, we obtain two additional criteria for uniform rectifiability. One is given in terms of the so-called $S<N$ estimates, and another in terms of a suitable corona decomposition involving $L$-harmonic and $L^*$-harmonic measures. We also prove that if $L$-harmonic measure and $L^*$-harmonic measure satisfy a weak $A_\infty$-type condition, then $\partial Ω$ is $n$-uniformly rectifiable. In the process we obtain a version of Alt-Caffarelli-Friedman monotonicity formula for a fairly wide class of elliptic operators which is of independent interest and plays a fundamental role in our arguments.

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arXiv2016

Tangents, rectifiability, and corkscrew domains

In a recent paper, Csörnyei and Wilson prove that curves in Euclidean space of $σ$-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 $Σ\subseteq \mathbb{R}^{d+1}$ has the property that each ball centered on $Σ$ contains two large balls in different components of $Σ^{c}$ and $Σ$ has $σ$-finite $\mathscr{H}^{d}$-measure, then it has $d$-dimensional tangent points in a set of positive $\mathscr{H}^{d}$-measure. We also give shorter proofs that Semmes surfaces are uniformly rectifiable and, if $Ω\subseteq \mathbb{R}^{d+1}$ is an exterior corkscrew domain whose boundary has locally finite $\mathscr{H}^{d}$-measure, one can find a Lipschitz subdomain intersecting a large portion of the boundary.

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