arXiv subjects
Publications and source records attributed to Jonas Azzam.
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.
Garnett, Killip, and Schul have exhibited a doubling measure $μ$ with support equal to $\mathbb{R}^{d}$ which is $1$-rectifiable, meaning there are countably many curves $Γ_{i}$ of finite length for which $μ(\mathbb{R}^{d}\backslash \bigcup Γ_{i})=0$. In this note, we characterize when a doubling measure $μ$ with support equal to a connected metric space $X$ has a $1$-rectifiable subset of positive measure and show this set coincides up to a set of $μ$-measure zero with the set of $x\in X$ for which $\liminf_{r\rightarrow 0} μ(B_{X}(x,r))/r>0$.
We study absolute continuity of harmonic measure with respect to surface measure on domains $Ω$ that have large complements. We show that if $Γ\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $ \mathbb{R}^{d+1}$ into two NTA domains then $ω_Ω\ll \mathscr{H}^{d}$ on $Γ\cap \partialΩ$. This result is a natural generalisation of a result of Wu in [Wu86]. We also prove that almost every point in $Γ\cap\partialΩ$ is a cone point if $Γ$ is a Lipschitz graph. Combining these results and a result from [AHMMMTV], we characterize sets of absolute continuity with finite $\mathscr{H}^{d}$-measure both in terms of the cone point condition and in terms of the rectifiable structure of the boundary. This generalizes the results of McMillan in [McM69] and Pommerenke in [Pom86]. 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.
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 $Ω\subset \mathbb{R}^{d+1}$ with $\mathcal{H}^{d}(\partialΩ)<\infty$ and a subset $E\subset \partial Ω$ with positive harmonic measure yet zero $\mathcal{H}^{d}$-measure. In particular, this implies that a classical theorem of F. and M. Riesz theorem fails in higher dimensions for this type of domains.
We show that for uniform domains $Ω\subseteq \mathbb{R}^{d+1}$ whose boundaries satisfy a certain nondegeneracy condition that harmonic measure cannot be mutually absolutely continuous with respect to $α$-dimensional Hausdorff measure unless $α\leq d$. We employ a lemma that shows that at almost every nondegenerate point, we may find a tangent measure of harmonic measure whose support is the boundary of yet another uniform domain whose harmonic measure resembles the tangent measure.
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 if $Ω\subset\mathbb R^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $\textrm{VMO}(σ)$, where $σ$ is the surface measure of $Ω$, then the outer unit normal to $\partialΩ$ belongs to $\textrm{VMO}(σ)$ too. The analogous result fails for dimensions larger than $3$. This answers a question posed by Kenig and Toro and also by Bortz and Hofmann.
We show that if $Ω$ is an NTA domain with harmonic measure $w$ and $E\subseteq \partialΩ$ is contained in an Ahlfors regular set, then $w|_{E}\ll \mathscr{H}^{d}|_{E}$. Moreover, this holds quantitatively in the sense that for all $τ>0$ $w$ obeys an $A_{\infty}$-type condition with respect to $\mathscr{H}^{d}|_{E'}$, where $E'\subseteq E$ is so that $w(E\backslash E')<τw(E)$, even though $\partialΩ$ may not even be locally $\mathscr{H}^{d}$-finite. We also show that, for uniform domains with uniform complements, if $E\subseteq\partialΩ$ is the Lipschitz image of a subset of $\mathbb{R}^{d}$, then there is $E'\subseteq E$ with $\mathscr{H}^{d}(E\backslash E')<τ\mathscr{H}^{d}(E)$ upon which a similar $A_{\infty}$-type condition holds.
In the present paper we prove that for any open connected set $Ω\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial Ω$ with $\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic measure $ω$ with respect to the Hausdorff measure on $E$ implies that $ω|_E$ is rectifiable. This solves an open problem on harmonic measure which turns out to be an old conjecture even in the planar case $n=1$.
In the present paper we sketch the proof of the fact that for any open connected set $Ω\subset\mathbb{R}^{n+1}$, $n\geq 1$, and any $E\subset \partial Ω$ with $0<\mathcal{H}^n(E)<\infty$, absolute continuity of the harmonic measure $ω$ with respect to the Hausdorff measure on $E$ implies that $ω|_E$ is rectifiable.
We show that if $n\geq 1$, $Ω\subset \mathbb R^{n+1}$ is a connected domain with porous boundary, and $E\subset \partialΩ$ is a set of finite and positive Hausdorff $H^{n}$-measure upon which the harmonic measure $ω$ is absolutely continuous with respect to $H^{n}$, then $ω|_E$ is concentrated on an $n$-rectifiable set.
A natural quantity that measures how well a map $f:\mathbb{R}^{d}\rightarrow \mathbb{R}^{D}$ is approximated by an affine transformation is \[ω_{f}(x,r)=\inf_{A}\left(\frac{1}{|B(x,r)|}\int_{B(x,r)}\left(\frac{|f-A|}{|A'|r}\right)^{2}\right)^{\frac{1}{2}},\] where the infimum ranges over all non constant affine transformations. This is natural insofar as it is invariant under rescaling $f$ in either its domain or image. We show that if $f:\mathbb{R}^{d}\rightarrow \mathbb{R}^{D}$ is quasisymmetric and its image has a sufficient amount of rectifiable structure (although not necessarily $\mathcal{H}^{d}$-finite), then $ω_{f}(x,r)^{2}\frac{dxdr}{r}$ is a Carleson measure on $\mathbb{R}^{d}\times(0,\infty)$. Moreover, this is an equivalence: the existence of such a Carleson measure implies that, in every ball $B(x,r)\subseteq \mathbb{R}^{d}$, there is a set $E$ occupying 90$%$ of $B(x,r)$, say, upon which $f$ is bi-Lipschitz (and hence guaranteeing rectifiable pieces in the image). En route, we make a minor adjustment to a theorem of Semmes to show that quasisymmetric maps of subsets of $\mathbb{R}^{d}$ into $\mathbb{R}^{d}$ are bi-Lipschitz on a large subset quantitatively.
We show that a Radon measure $μ$ 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 $μ$-almost everywhere. The converse of this result is proven in a companion paper by the second author, and hence these two results give a classification of all $n$-rectifiable measures which are absolutely continuous with respect to $H^{n}$. Further, in this paper we also investigate the relationship between the Jones' square function and the so called Menger curvature of a measure with linear growth.
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 $L^1$ Wasserstein distance. We show that measure satisfying certain self-similarity conditions admits a unique (up to multiplication by a constant) flat tangent measure at almost every point. This allows us to decompose the support into rectifiable pieces of various dimensions.
Let $μ$ be a doubling measure in $\mathbb{R}^n$. We investigate quantitative relations between the rectifiability of $μ$ and its distance to flat measures. More precisely, for $x$ in the support $Σ$ of $μ$ and $r > 0$, we introduce a number $α(x,r)\in (0,1]$ that measures, in terms of a variant of the $L^1$-Wasserstein distance, the minimal distance between the restriction of $μ$ to $B(x,r)$ and a multiple of the Lebesgue measure on an affine subspace that meets $B(x,r/2)$. We show that the set of points of $Σ$ where $\int_0^1 α(x,r) \frac{dr}{r} < \infty$ can be decomposed into rectifiable pieces of various dimensions. We obtain additional control on the pieces and the size of $μ$ when we assume that some Carleson measure estimates hold. Soit $μ$ une mesure doublante dans $\mathbb{R}^n$. On étudie des relations quantifiées entre la rectifiabilité de $μ$ et la distance entre $μ$ et les mesures plates. Plus précisément, on utilise une variante de la $L^1$-distance de Wasserstein pour définir, pour $x$ dans le support $Σ$ de $μ$ et $r>0$, un nombre $α(x,r)$ qui mesure la distance minimale entre la restriction de $μ$ à $B(x,r)$ et une mesure de Lebesgue sur un sous-espace affine passant par $B(x,r/2)$. On décompose l'ensemble des points $x\in Σ$ tels que $\int_0^1 α(x,r) \frac{dr}{r} < \infty$ en parties rectifiables de dimensions diverses, et on obtient un meilleur contrôle de ces parties et de la taille de $μ$ quand les $α(x,r)$ vérifient certaines conditions de Carleson.
We show that if $Ω\subset \mathbb{R}^{n+1}$, $n\geq 1$, 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 $Ω$ implies the existence of exterior Corkscrew points at all scales, so that in fact, $Ω$ is a chord-arc domain, i.e., a domain with an Ahlfors-David regular boundary which satisfies both interior and exterior Corkscrew conditions, and an interior Harnack Chain condition. We discuss some implications of this result, for theorems of F. and M. Riesz type, and for certain free boundary problems.
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 pieces of bi-Lipschitz images of ${\Bbb R}^n$. One main novelty of these results is that we analyze quasiplanes in arbitrary codimension $N-n$. To establish the big pieces criterion, we prove new extension theorems for "almost affine" maps, which are of independent interest. This work is related to investigations by Tukia and Väisälä on extensions of quasisymmetric maps with small distortion.