## Jonas Azzam

##### University of Edinburgh Lecturer

JCMB 4613, Kings Buildings
j.azzam$@$ed.ac.uk

### Brief CV

2011: University of California-Los Angeles, PhD

##### PhD Students
Jean Burnazyan
Matthew Hyde
Michele Villa (PhD 2020, now at U. Helsinki)

## Research

Rectifable Sets, Harmonic Measure, General Geometric Measure Theory, Miscellaneous

### Rectifiable Sets

Given a set (or metric space) how can I tell if and how well the set resembles Rd? There are various degrees of resemblance you could demand, like being isometric. One variant that I consider is the property of being rectifiable: a set is d-rectifiable if it can be covered up to d-dimensional measure zero by Lipschitz images of Rd. One can think of these sets as measure theoretic versions of smooth manifords: a rectifiable set is one that is "locally" parametrized by Lipschitz chart maps. There are even different degrees of rectifiability (like uniform rectifiability). Much of my work concerns with classifying when a set is rectifiable and studying the properties of rectifiable sets. There is a particular emphasis on quantitative analysis, and this only became more important around the 90s in connection with boundedness of singular integrals and Painleve's probelm.

We show that Ahlfors $d$-regular set $E$ in $\mathbb{R}^{n}$ is uniformly rectifiable if the set of pairs $(x,r)\in E\times (0,\infty)$ where $\mathscr{H}^{d}_{\infty}(E\cap B(x,r))<(2r)^{d}(1-\epsilon)$ is a Carleson set for every $\epsilon>0$. To prove this, we generalize a result of Schul by proving, if $X$ is a $C$-doubling metric space, $\epsilon,\rho\in (0,1)$, $A>1$, 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 $\sum \left\{r_{B}^{s}: B\in \mathscr{B}, \frac{\mathscr{H}^{s}_{\rho r_{B}}(X\cap AB)}{(2r_{B})^{s}}>1+\epsilon\right\} \lesssim_{C,A,\epsilon,\rho} \mathscr{H}^{s}(X).$ 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.
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.
Arxiv
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 $\beta_{E}(R)$ denotes the square sum of $\beta$-numbers over subcubes of $R$ as in the Traveling Salesman Theorem for higher dimensional sets [AS18], then $\mathscr{H}^{d}(R)+\sum_{Q\subseteq R\atop Q\in \mathscr{B}} ({\rm diam} Q)^{d}\sim \beta_{E}(R).$ 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.
Arxiv.
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 $d$-dimensional Lipschitz graphs and $\mu \ll \mathscr{H}^{d}$. The characterization is in terms of the a.e. finiteness Jones function involving the so-called $\alpha$-numbers. We also give an example of an unrectifiable measure that is not pointwise doubling for which the Jones function is finite almost everywhere. This answers a question left open in a former work by Azzam, David, and Toro. Slides

Epilogue: The counterexample mentioned above means that we cannot characterize rectifiable measures using $\alpha$-numbers we define. However, Damian Dąbrowski has developed a full characterization using $\alpha$-numbers based on the $W_{2}$-Wasserstein distance.

We develop a quantity $\beta_{E}^{d}(Q)$ that measures the $d$-flatness of a set $E\subseteq \mathbb{R}^{n}$ inside a cube $Q$ so that, if $E$ is a Reifenberg flat surface, $(\mbox{diam } E)^{d}+\sum_{Q\subseteq \mathbb{R}^{n} \atop Q\cap E\neq\emptyset} \beta_{E}(3Q)^{2}\ell(Q)^{d}\sim_{n} \mathscr{H}^{d}(E).$ The result also holds for surfaces $E\subseteq \mathbb{R}^{n}$ satisfying Condition B, that is, every ball centered on $E$ has two large balls in different components of $E^{c}$. This gives a generalization of the (Euclidean) Analyst's Traveling Salesman Theorem of Peter Jones and Kate Okikiolu, which originally held only for curves.

Note: The published version has many annoying typos and small mistakes that make for hard reading. For this reason, I am keeping an up-to-date version here where I correct any mistakes that I happen to find or that people point out. If you have any questions about the paper or notice an error, let me know so I can sort it. Slides, Video.

In a recent paper, Csörnyei 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 the boundary of a corkscrew domain in $\mathbb{R}^{d+1}$ has $\sigma$-finite $\mathscr{H}^{d}$-measure, then it has $d$-dimensional tangent points in a set of positive $d$-measure. We also give a simpler proof of the well-known fact that, if $\Omega\subseteq \mathbb{R}^{d+1}$ 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.
Arxiv.
Let $\mu$ be a doubling measure in $\mathbb{R}^n$. We introduce two parts of the support of $\mu$ where $\mu$ has some quantitative self-similarity properties, measured with a variant of the $L^1$ Wasserstein distance, and prove that at each point of these two parts, all the tangent measures to $\mu$ are multiples of some flat measure (the Lebesgue measure on a vector subspace). We use this to decompose the two parts into rectifiable pieces of various dimensions.
Arxiv.
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-zero affine transformations $A$. 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 $\omega_{f}(x,r)^{2}\frac{dxdr}{r}$ is a Carleson measure on $\mathbb{R}^{d}\times(0,\infty)$. Moreover, this is an equivalence: if this is a Carleson measure, then, 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.
Arxiv., Slides.
Garnett, Killip, and Schul have exhibited a doubling measure $\mu$ with support equal to $\mathbb{R}^{d}$, $d\geq 2$, 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 give a characterisation of when a doubling measure with connected support in a metric space has a $1$-rectifiable subset of positive measure and show this set coincides up to a set of $\mu$-measure zero with the set of $x\in \mbox{supp} \mu$ for which $\liminf_{r\rightarrow 0} \mu(B(x,r))/r>0$.
Arxiv., Slides.
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 minimal distance between the restriction of $\mu$ 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 $\Sigma$ where $\int_0^1 \alpha(x,r) {dr \over r} < \infty$ can be decomposed into rectifiable pieces of various dimensions. We obtain additional control on the pieces and the size of $\mu$ when we assume that some Carleson measure estimates hold.
Arxiv..
We generalize the results of David and Semmes by providing sufficient $\beta$-number-type conditions for rectifiability of Radon measures, requiring only that their upper densities are positive and finite almost everywhere. More precisely, let $n>0$ and let $\mu$ be a Radon measure in $\mathbb{R}^{d}$. Suppose that for some $1\leq n\neq d$ we have $\theta^{n,*}(\mu,x)\in(0,\infty)$ $\mu$-a.e. and that $\int_{0}^{1}\beta_{\mu,2}(x,r)^{2}\frac{dr}{r}<\infty\;\; \mu-a.e.$ where $\beta_{\mu,2}(x,r)^{2}=\inf_{L\in G(n,d)}\int_{B(x,r)}\left(\frac{dist(x,L)}{r}\right)^{2}\frac{dr}{r^{n}}$ and $G(n,d)$ denotes the set of $n$-dimensional planes in $\mathbb{R}^{d}$. Then $\mu$ may be covered up to a set of $\mu$-measure zero by Lipschitz images. Combined with a forthcoming paper by Tolsa, this $\beta$-number condition completely characterizes $n$-rectifiable Radon measures with positive and finite upper $n$-densities.
Arxiv..

Epilogue: This result was generalized by Edelen, Naber and Valtorta. They have been used in a vareity of contexts, like line defects in liquid quasicrystals, obstacle problems, calculus of variations, and singular integrals.
A quasiplane $f(V)$ is the image of an $n$-dimensional Euclidean subspace $V$ of $\mathbb{R}^N$ ($1\leq n\leq N-1$) under a quasiconformal map $f:\mathbb{R}^N\rightarrow\mathbb{R}^N$ . 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 $\mathbb{R}^n$. 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.
Arxiv..
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 approximately a seminorm. This is a quantitative version of Kirchheim's metric differentiation result from 1994. Our result is in the form of a Carleson packing condition.
Article..
All Lipschitz maps from $R^7$ to $R^3$ are orthogonal projections". This is of course quite false as stated. It turns out however, that there is a surprising grain of truth in this statement. We prove a global implicit function theorem. In particular we show that any Lipschitz map $f:\mathbb{R}^n\times \mathbb{R}^m\to\mathbb{R}^n$ (with $n$-dim. image) can be precomposed with a bi-Lipschitz map $\bar{g}:\mathbb{R}^n\times \mathbb{R}^m\to \mathbb{R}^n\times \mathbb{R}^m$ such that $f\circ \bar{g}$ will satisfy, when we restrict to a large portion of the domain $E\subset \mathbb{R}^n\times \mathbb{R}^m$, that $f\circ \bar{g}$ is bi-Lipschitz in the first coordinate, and constant in the second coordinate. Geometrically speaking, the map $\bar{g}$ distorts $\mathbb{R}^{n+m}$ in a controlled manner so that the fibers of $f$ are straightened out. Furthermore, our results stay valid when the target space is replaced by any metric space. A main point is that our results are quantitative: the size of the set $E$ on which behavior is good is a significant part of the discussion. Our estimates are motivated by examples such as Kaufman's 1979 construction of a $C^1$ map from $[0,1]^3$ onto $[0,1]^2$ with rank $\leq 1$ everywhere.

On route we prove an extension theorem which is of independent interest. We show that for any $D\geq n$, any Lipschitz function $f:[0,1]^n\to \mathbb{R}^D$ gives rise to a large (in an appropriate sense) subset $E\subset [0,1]^n$ such that $f|_E$ is bi-Lipschitz and may be extended to a bi-Lipschitz function defined on all of $\mathbb{R}^n$. This extends results of P. Jones and G. David, from 1988. As a simple corollary, we show that $n$-dimensional Ahlfors-David regular spaces lying in $\mathbb{R}^{D}$ having big pieces of bi-Lipschitz images also have big pieces of big pieces of Lipschitz graphs in $\mathbb{R}^{D}$. This was previously known only for $D\geq 2n+1$ by a result of G. David and S. Semmes.
Arxiv, Slides.
Epilogue:This result was the main part of my thesis. It was improved upon by Raanan Schul and Guy C. David in 2020, see here for video seminar about their result, and the video also starts by giving a synopsis of our result.
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 in $\tilde\Gamma$, which has length bounded by a fixed constant multiple of the Euclidean distance between $x$ and $y$. Thus, for any set $K$ in $d-$dimensional Euclidean space we have a set $\tilde\Gamma$ as above such that $\tilde\Gamma$ has comparable Hausdorff length to a shortest connected set containing $K$. Constants appearing here depend only on the ambient dimension $d$. In the case where $\Gamma$ is Reifenberg flat, our constants are also independent the dimension $d$, and in this case, our theorem holds for $\Gamma$ in an infinite dimensional Hilbert space. This work closely related to $k-$spanners, which appear in computer science.
Arxiv, Slides.

### Harmonic Measure

For a domain $\Omega\subseteq \mathbb{R}^{n}$, its harmonic measure with pole at $x$, denoted $\omega_{\Omega}^{x}$ can be defined two ways: one is that, for $A\subseteq \partial\Omega$, $\omega_{\Omega}^{x}(A)$ is the probability that a Brownian motion starting at $x\in \Omega$ first hits $\partial \Omega$ in the set $A$. Alternatively (and assuming our domain is "nice" enough), for each continuous function $f$ one can solve the Dirichlet problem to find a harmonic function $u_{f}$ on $\Omega$ that agrees with $f$ on the boundary; for $x\in \Omega$, the map $f\mapsto u_{f}(x)$ is a bounded linear functional and the Riesz Representation Theorem supplies a measure $\omega_{\Omega}^{x}$ so that $u_{f}(x) = \int_{\Omega}fd\omega_{\Omega}^{x}$. This is quite a natural measure to give the boundary of a domain, and so it's natural to ask what the relationship between the geometry of the domain and the behavior of its harmonic measure is. My work in this area has mostly focused on the relationship between the reguarity of harmonic measure and the rectifiable structure of the boundary.

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 generalized Jones $\beta$-numbers is finite. Using this, we give estimates on the fluctuation of Green's function in a uniform domain in terms of the $\beta$-numbers. As a corollary, for bounded NTA domains , if $B_{\Omega}=B(x_{\Omega},c\mathrm{diam} \Omega)$ is so that $2B_{\Omega}\subseteq \Omega$, we obtain that $(\mathrm{diam} \partial\Omega)^{d} + \int_{\Omega\backslash B_{\Omega}} \left|\frac{\nabla^2 G_{\Omega}(x_{\Omega},x)}{G_{\Omega}(x_{\Omega},x)}\right|^{2} \mathrm{dist}(x,\Omega^c)^{3} dx \sim \mathscr{H}^{d}(\partial\Omega).$ Secondly, we also use $\beta$-numbers to estimate how much harmonic measure fails to be $A_{\infty}$-weight for semi-uniform domains with Ahlfors regular boundaries.
Slides
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 $s$.
Article.
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 in the nature of best possible: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds).
Arxiv.
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 firstly remove the John condition from their result, and secondly, we show 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. This requires finding suitable substitutes for classical estimates for nontangentially accessible domains that work in semi-uniform domains with the capacity density condition, which may be of independent interest.
Arxiv.
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 $\Omega_1$ 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 our result is that the harmonic measures do not need to satisfy any doubling condition.
Arxiv.
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 $\Omega\subseteq \mathbb{R}^{n+1}$:
1. We extend the results of Kenig, Preiss, and Toro [KPT09] by showing mutual absolute continuity of interior and exterior elliptic measures for 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 $\omega$ 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.
Arxiv.
Let $\Omega\subseteq \mathbb{R}^{n+1}$, $n\geq 2$, 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 $\partial\Omega$ is uniformly $n$-rectifiable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^∗v=0$ in $\Omega$ is $\epsilon$-approximable if and only if every bounded solution of $Lu=0$ and every bounded solution of $L^∗v=0$ in $\Omega$ 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 $\partial\Omega$ 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.
Arxiv.
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 natural generalisation of a result of Wu in [Wu86]. We also prove that almost every point in $\Gamma\cap\partial\Omega$ is a cone point if $\Gamma$ 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.
Arxiv, Slides.
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.
Arxiv.
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 original domain $\Omega$. Our main application is to obtain bounds on $s$ when harmonic measure has positive and finite upper and lower $s$-densities on a set of positive harmonic measure. In the special case that $\Omega$ is nontangentially accessible, our bounds imply that $s=d$.
Arxiv.

Epilogue: Tolsa recently generalized the results of this paper by showing that harmonic measure for any domain whose boundary has positive and finite $s$-dimensional measure is singular with respect to the $s$-dimensional surface measure on the boundary.
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.
Arxiv, Slides.
For $d>2$, we show there exists locally Reifenberg flat domains $\Omega\subseteq \mathbb{R}^{d+1}$ with $\mathscr{H}^{d}(\partial\Omega)$<$\infty$ and a subset $E\subseteq \partial\Omega$ with positive harmonic measure yet zero $\mathscr{H}^{d}$-measure. This generalizes a counter-example of Wu and answers a conjecture of Badger. The main mart of the proof consists in showing how, for a prescribed compact subset $E$ of $\partial \Omega$, where $\Omega$ is a Reifenberg flat domain, to construct a larger domain $\Omega^{+}$ whose boundary is also Reifenberg flat and intersects $\partial\Omega$ on $E$.
Arxiv.
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 dimensions larger than $3$. This answers a question posed by Kenig and Toro and also by Bortz and Hofmann.
Arxiv.
We show that if $\Omega \subseteq \mathbb{R}^{n+1}$, $n\geq 1$, 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 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.
Arxiv.
If $\Omega\subseteq\mathbb{R}^{d+1}$ is an NTA domain with harmonic measure $w$ and $E\subseteq \partial\Omega$ is contained in an Ahlfors regular set, then for all $\tau>0$ there is $E'\subseteq E$ $d$-rectifiable with $w(E\backslash E')$<$\tau w(E)$ and $w|_{E'}\ll \mathscr{H}^{d}|_{E'}\ll w|_{E'}$, so in particular, $w|_{E}\ll \mathscr{H}^{d}|_{E}$. Moreover, this holds quantitatively in the sense that $w$ obeys an $A_{\infty}$-type condition with respect to $\mathscr{H}^{d}$ on $E'$, even though $\partial\Omega$ may not even be locally $\mathscr{H}^{d}$-finite. We also show that if $(\Omega^{c})^{\circ}$ is also NTA, and if $E\subseteq\partial \Omega$ is in a Lipschitz image of $\mathbb{R}^{d}$, then for all $\tau>0$ there is $E'\subseteq E$ with $\mathscr{H}^{d}(E\backslash E')$<$\tau\mathscr{H}^{d}(E)$, $w|_{E'}\ll\mathscr{H}^{d}|_{E'}\ll w|_{E'}$, and such that a similar $A_{\infty}$-condition holds.
Arxiv, Slides.
We show that if $n\geq 1$, $\Omega\subset \mathbb{R}^{n+1}$ is a connected domain, and $E\subset \partial\Omega$ is a set of finite and positive $\mathscr{H}^{n}$-measure upon which the harmonic measure $\omega$ is absolutely continuous with respect to $\mathscr{H}^{n}$, then $\omega|_E$ is $n$-rectifiable set.
Arxiv, Slides

### General Geometric Measure Theory

Most of my work falls into the broad category of geometric measure theory. Here I list a few one-off results in this area that aren't connected to my more specific subareas areas.

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 Koskela, Nandi, and Nicolau with John domains in place of chord-arc domains when $n=2$, $s=1$, and $\Omega$ is a simply connected planar domain. As an application, our result combined with works of Koskela and Lehrbäck give a characterization of which domains support pointwise $(p,\beta)$-Hardy inequalities for $\beta$<$p-1$.
Arxiv.
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.
Arxiv.
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 $r\leq r_{0}$, then ${\rm dim} X\geq 1+c\beta^{2}$. This generalizes a theorem of Bishop and Jones (which was originally for Euclidean spaces) and another of by Bishop and Tyson about the dimension of antenna sets.
Arxiv.

### Miscellaneous

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 known uniqueness results for the 2D Euler equations in $L^1 \cap BMO$ and provide a mild improvement to the recent results of Rusin for the 2D inviscid surface quasi-geostrophic (SQG) equations, which are now to our knowledge, the best results known for this model. We also obtain what are (to our knowledge) the strongest known uniqueness results for weak solutions to the Vlasov-Poisson equations, extending the results of Robert and Loeper to include densities in $L^1 \cap BMO$, and the strongest known uniqueness results for the Patlak-Keller-Segel models. We obtain these results via technical refinements of energy methods which are well-known in the $L^2$ setting but are less well-known in the $\dot{H}^{-1}$ setting. The $\dot{H}^{-1}$ method can be considered a generalization of Yudovich's classical method and is naturally applied to equations such as the Patlak-Keller-Segel models with nonlinear diffusion, and other variants and the Vlasov-Poisson equation and other kinetic models. The $\dot{H}^{-1}$ method is a variant of the classical method due to Yudovich for the 2D Euler equations. An important tool in our analysis is a Sobolev embedding lemma which shows that velocity fields $v$ with $\nabla v \in BMO$ are locally log-Lipschitz. Hence, they satisfy the Osgood uniqueness criterion for ODE, which is known to play an important role in the uniqueness of active scalars.
Arxiv.