Pertti Mattila
This is a survey on rectifiability. I discuss basic properties of rectifiable sets, measures, currents and varifolds and their role in complex and harmonic analysis, potential theory, calculus of variations, PDEs and some other topics.
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In this note we show that for every measurable function on $\mathbb{R}^n$ the set of points where the blowup exists and is not constant is $(n-1)$-rectifiable. In particular, for every $u\in L^1_{loc}(\mathbb{R}^n)$ the jump set $J_u$ is $(n-1)$-rectifiable.
A Radon measure $μ$ is $n$-rectifiable if $μ\ll\mathcal{H}^n$ and $μ$-almost all of $\text{supp}\,μ$ can be covered by Lipschitz images of $\mathbb{R}^n$. In this paper we give a necessary condition for rectifiability in terms of the so-called $α_2$ numbers -- coefficients quantifying flatness using Wasserstein distance $W_2$. In a recent article we showed that the same condition is also sufficient for rectifiability, and so we get a new characterization of rectifiable measures.
Rectifiability of flat chains
我们证明（不使用Federer的结构定理），有限质量平坦链在任何系数群是可求长的当且仅当几乎所有的0维切片是可求长的。这意味着每一个有限质量和有限尺寸的平面链都是可求长的。它还导致了一个简单的充要条件的系数群，以使每一个有限质量的扁平链是可求长。
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…
We present the converse to a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, for $n\geq 2$, for an ADR domain $Ω\subset \re^{n+1}$ which satisfies the Harnack Chain condition plus an interior (but not exterior) Corkscrew condition, we show that absolute continuity of harmonic measure with respect to surface measure on $\partialΩ$, with scale invariant higher integrability of the Poisson kernel, is sufficient to imply uniformly rectifiable of $\partialΩ$.
In geometric measure theory, there is interest in studying the interaction of measures with rectifiable sets. Here, we extend a theorem of Badger and Schul in Euclidean space to characterize rectifiable pointwise doubling measures in Hilbert space. Given a measure $μ$, we construct a multiresolution family $\mathscr{C}^μ$ of windows, and then we use a weighted Jones' function $\hat{J}_2(μ, x)$ to record how well lines approximate the distribution of mass in each window. We show that when $μ$ is rectifiable, the mass is sufficiently concentrated around a lines at each scale and that the convers…
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.
We present a higher dimensional, scale-invariant version of a classical theorem of F. and M. Riesz. More precisely, we establish scale invariant absolute continuity of harmonic measure with respect to surface measure, along with higher integrability of the Poisson kernel, for a domain $Ω\subset \mathbb{R}^{n+1},\, n\geq 2$, with a uniformly rectifiable boundary, which satisfies the Harnack Chain condition plus an interior (but not exterior) corkscrew condition. In a companion paper to this one [HMU], we also establish a converse, in which we deduce uniform rectifiability of the boundary, assum…
Let $E\subset \mathbb{R}^{n+1}$, $n\ge 2$, be a closed, Ahlfors-David regular set of dimension $n$ satisfying the "Riesz Transform bound" $$\sup_{\varepsilon>0}\int_E\left|\int_{\{y\in E:|x-y|>\varepsilon\}}\frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)\right|^2 dH^n(x) \leq C \int_E|f|^2 dH^n .$$ Assume further that $E$ is the boundary of a domain $Ω\subset \mathbb{R}^{n+1}$ satisfying the Harnack Chain condition plus an interior (but not exterior) Corkscrew condition. Then $E$ is uniformly rectifiable.
In the late `90s there was a flurry of activity relating $1$-rectifiable sets, boundedness of singular integral operators, the analytic capacity of a set, and the integral Menger curvature in the plane. In `99 Leger extended the results for Menger curvature to $1$-rectifiable sets in higher dimension, as well as to the codimension one case. A decade later, Lerman and Whitehouse, and later Meurer, found higher-dimensional geometrically motivated generalizations of Menger curvature that yield results about the uniform rectifiability of measures and the rectifiability of sets respectively. In thi…
Aim of this paper is a finer analysis of the group of flat chains with coefficients in $Z_p$ introduced in a recent paper by Ambrosio-Katz, by taking quotients of the group of integer rectifiable currents, along the lines of the the Ziemer and Federer approach. We investigate the typical questions of the theory of currents, namely rectifiability of the measure-theoretic support and boundary rectifiability. Our main result can also be interpreted as a closure theorem for the class of integer rectifiable currents with respect to a (much) weaker convergence, induced by flat distance mod $p$, and …
Our purpose is to state quantitative conditions ensuring the rectifiability of a $d$--varifold $V$ obtained as the limit of a sequence of $d$--varifolds $(V_i)_i$ which need not to be rectifiable. More specifically, we introduce a sequence $\left\lbrace \mathcal{E}_i \right\rbrace_i$ of functionals defined on $d$--varifolds, such that if $\displaystyle \sup_i \mathcal{E}_i (V_i) < +\infty$ and $V_i$ satisfies a uniform density estimate at some scale $β_i$, then $V = \lim_i V_i$ is $d$--rectifiable. \noindent The main motivation of this work is to set up a theoretical framework where curves, su…
On Singular Integrals and Quantitative Rectifiability in Parabolic Space and the Heisenberg Group
本文将David和Semmes关于奇异积分与定量可测性的定理扩展到抛物空间和海森堡群。
In this paper it is shown that if $μ$ is an n-dimensional Ahlfors-David regular measure in $R^d$ which satisfies the so-called weak constant density condition, then $μ$ is uniformly rectifiable. This had already been proved by David and Semmes in the cases n=1, 2 and d-1, and it was an open problem for other values of n. The proof of this result relies on the study of the n-uniform measures in $R^d$. In particular, it is shown here that they satisfy the "big pieces of Lipschitz graphs" property.
For a set $E$ of positive and finite length, we prove that if the Huovinen transform (the convolution operator with kernel $z^k/|z|^{k+1}$ for an odd number $k$) associated to $E$ exists in principal value, then $E$ is rectifiable.
For a Radon measure $μ$ on $\mathbb{R}^d$, define $C^n_μ(x, t)= \ (\frac{1}{t^n} \ |\int_{B(x,t)} \frac{x-y}{t} \, dμ(y)\ | \ )$. This coefficient quantifies how symmetric the measure $μ$ is by comparing the center of mass at a given scale and location to the actual center of the ball. We show that if $μ$ is $n$-rectifiable, then $ \int_0^\infty |C^n_μ(x,t)|^2 \frac{dt}{t} < \infty \, \, μ\mbox{-almost everywhere}. $ Together with a previous result of Mayboroda and Volberg, where they showed that the converse holds true, this gives a characterisation of $n$-rectifiability. To prove our main re…
We consider second order divergence form elliptic operators with $W^{1,1}$ coefficients, in a uniform domain $Ω$ with Ahlfors regular boundary. We show that the $A_\infty$ property of the elliptic measure associated to any such operator implies that $Ω$ is a set of locally finite perimeter whose boundary, $\partialΩ$, is rectifiable. As a corollary we show that for this type of operators, absolute continuity of the surface measure with respect to the elliptic measure is enough to guarantee rectifiability of the boundary. In the case that the coefficients are continuous we obtain additional inf…
In this paper we characterize the so called uniformly rectifiable sets of David and Semmes in terms of the Wasserstein distance $W_2$ from optimal mass transport. To obtain this result, we first prove a localization theorem for the distance $W_2$ which asserts that if $μ$ and $ν$ are probability measures in $R^n$, $ϕ$ is a radial bump function smooth enough so that $\intϕdμ\gtrsim1$, and $μ$ has a density bounded from above and from below on the support of ϕ, then $W_2(ϕμ,aϕν)\leq c W_2(μ,ν),$ where $a=\intϕdμ/ \intϕ\,dν$.
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 $n=1$.
We show that, given a set $E\subset \mathbb R^{n+1}$ with finite $n$-Hausdorff measure $H^n$, if the $n$-dimensional Riesz transform $$R_{H^n|E} f(x) = \int_{E} \frac{x-y}{|x-y|^{n+1}} f(y) dH^n(y)$$ is bounded in $L^2(H^n|E)$, then $E$ is $n$-rectifiable. From this result we deduce that a compact set $E\subset\mathbb R^{n+1}$ with $H^n(E)<\infty$ is removable for Lipschitz harmonic functions if and only if it is purely $n$-unrectifiable, thus proving the analog of Vitushkin's conjecture in higher dimensions.
We obtain Dini conditions with "exponent 2" that guarantee that an asymptotically conformal quasisphere is rectifiable. In particular, we show that for any e>0 integrability of (esssup_{1-t < |x| < 1+t} K_f(x)-1)^{2-e} dt/t implies that the image of the unit sphere under a global quasiconformal homeomorphism f is rectifiable. We also establish estimates for the weak quasisymmetry constant of a global K-quasiconformal map in neighborhoods with maximal dilatation close to 1.
We construct extensions of Varopolous type for functions $f \in \text{BMO}(E)$, for any uniformly rectifiable set $E$ of codimension one. More precisely, let $Ω\subset \mathbb{R}^{n+1}$ be an open set satisfying the corkscrew condition, with an $n$-dimensional uniformly rectifiable boundary $\partial Ω$, and let $σ:= \mathcal{H}^n\lfloor_{\partial Ω}$ denote the surface measure on $\partial Ω$. We show that if $f \in \text{BMO}(\partial Ω,dσ)$ with compact support on $\partial Ω$, then there exists a smooth function $V$ in $Ω$ such that $|\nabla V(Y)| \, dY$ is a Carleson measure with Carleson…
This paper is devoted to show that the flatness of tangents of $1$-codimensional measures in Carnot Groups implies $C^1_\mathbb{G}$-rectifiability. As applications we prove that measures with $(2n+1)$-density in the Heisenberg groups $\mathbb{H}^n$ are $C^1_{\mathbb{H}^n}$-rectifiable, providing the first non-Euclidean extension of Preiss's rectifiability theorem and a criterion for intrinsic Lipschitz rectifiability of finite perimeter sets in general Carnot groups.
Ricci curvature bounded below and uniform rectifiability
本文证明Ahlfors正则RCD空间及其边界的一致可求长性，并由此得到Lipschitz函数的定量微分结果。
A well known notion of $k$-rectifiable set can be formulated in any metric space using Lipschitz images of subsets of $\mathbb{R}^k$. We prove some characterizations of $k$-rectifiability, when the metric space is an arbitrary homogeneous group. In particular, we show that the a.e. existence of the $(k,\mathbb{G})$-approximate tangent group implies $k$-rectifiability.
The main goal of this paper is to develop a concept of approximate differentiability of higher order for subsets of the Euclidean space that allows to characterize higher order rectifiable sets, extending somehow well known facts for functions. We emphasize that for every subset $ A $ of the Euclidean space and for every integer $ k \geq 2 $ we introduce the approximate differential of order $ k $ of $ A $ and we prove it is a Borel map whose domain is a (possibly empty) Borel set. This concept could be helpful to deal with higher order rectifiable sets in applications.
One goal of geometric measure theory is to understand how measures in the plane or higher dimensional Euclidean space interact with families of lower dimensional sets. An important dichotomy arises between the class of rectifiable measures, which give full measure to a countable union of the lower dimensional sets, and the class of purely unrectifiable measures, which assign measure zero to each distinguished set. There are several commonly used definitions of rectifiable and purely unrectifiable measures in the literature (using different families of lower dimensional sets such as Lipschitz i…
We prove a structure theorem for any $n$-rectifiable set $E\subset \mathbb{R}^{n+1}$, $n\ge 1$, satisfying a weak version of the lower ADR condition, and having locally finite $H^n$ ($n$-dimensional Hausdorff) measure. Namely, that $H^n$-almost all of $E$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\mathbb{R}^{n+1}\setminus E$. As a consequence, for harmonic measure in the complement of such a set $E$, we establish a non-degeneracy condition which amounts to saying that $H^n|_E$ is "absolutely continuous" with respect to harmonic measure in the …
We say a measure is $C^{1,α}$ $d$-rectifiable if there is a countable union of $C^{1,α}$ $d$-surfaces whose complement has measure zero. We provide sufficient conditions for a Radon measure in $\mathbb{R}^n$ to be $C^{1,α}$ $d$-rectifiable, with $α\in (0,1]$. The conditions involve a Bishop-Jones type square function and all statements are quantitative in that the $C^{1,α}$ constants depend on such a function. Along the way we also give sufficient conditions for $C^{1,α}$ parametrizations for Reifenberg flat sets in terms of the same square function. Key tools for the proof come from David and…
Let $Ω\subset\mathbb R^{n+1}$, $n\geq1$, be a corkscrew domain with Ahlfors-David regular boundary. In this paper we prove that $\partialΩ$ is uniformly $n$-rectifiable if every bounded harmonic function on $Ω$ is $\varepsilon$-approximable or if every bounded harmonic function on $Ω$ satisfies a suitable square-function Carleson measure estimate. In particular, this applies to the case when $Ω=\mathbb R^{n+1}\setminus E$ and $E$ is Ahlfors-David regular. Our results solve a conjecture posed by Hofmann, Martell, and Mayboroda in a recent work where they proved the converse statements. Here we …
Whitney extension theorem and intrinsic rectifiability of jump sets in Carnot-Carathéodory spaces
证明等正则Carnot-Carathéodory空间中局部可积函数的内在跳跃集具有内在可数可求长性，关键新工具是内在C^1函数的局部标量Whitney延拓定理。
We repurpose tools from the theory of quantitative rectifiability to study the qualitative rectifiability of measures in $\Bbb{R}^n$, $n\geq 2$. To each locally finite Borel measure $μ$, we associate a function $\widetilde J_2(μ, x)$ which uses a weighted sum to record how closely the mass of $μ$ is concentrated on a line in the triples of dyadic cubes containing $x$. We show that $\widetilde J_2(μ, x) < \infty$ $μ$-a.e. is a necessary condition for $μ$ to give full mass to a countable family of rectifiable curves. This confirms a conjecture of Peter Jones from 2000. A novelty of this result i…
Let $A(\cdot)$ be an $(n+1)\times (n+1)$ uniformly elliptic matrix with Hölder continuous real coefficients and let $\mathcal E_A(x,y)$ be the fundamental solution of the PDE $\mathrm{div} A(\cdot) \nabla u =0$ in $\mathbb R^{n+1}$. Let $μ$ be a compactly supported $n$-AD-regular measure in $\mathbb R^{n+1}$ and consider the associated operator $$T_μf(x) = \int \nabla_x\mathcal E_A(x,y)\,f(y)\,dμ(y).$$ We show that if $T_μ$ is bounded in $L^2(μ)$, then $μ$ is uniformly $n$-rectifiable. This extends the solution of the codimension $1$ David-Semmes problem for the Riesz transform to the gradient…
Let $E$ be a set in $\mathbb R^d$ with finite $n$-dimensional Hausdorff measure $H^n$ such that $\liminf_{r\to0}r^{-n} H^n(B(x,r)\cap E)>0$ for $H^n$-a.e. $x\in E$. In this paper it is shown that $E$ is $n$-rectifiable if and only if $$\int_0^1 \left|\frac{H^n(B(x,r)\cap E)}{r^n} - \frac{H^n(B(x,2r)\cap E)}{(2r)^n}\right|^2\,\frac{dr}r < \infty$$ for $H^n$-a.e. $x\in E$; and also if and only if $$ \lim_{r\to0}\left(\frac{H^n(B(x,r)\cap E)}{r^n} - \frac{H^n(B(x,2r)\cap E)}{(2r)^n}\right) = 0$$ for $H^n$-a.e. $x\in E$. Other more general results involving Radon measures are also proved.
In this paper it is shown that an Ahlfors-David $n$-dimensional measure $μ$ on $\mathbb{R}^d$ is uniformly $n$-rectifiable if and only if for any ball $B(x_0,R)$ centered at $\operatorname{supp}(μ)$, $$ \int_0^R \int_{x\in B(x_0,R)} \left|\frac{μ(B(x,r))}{r^n} - \frac{μ(B(x,2r))}{(2r)^n} \right|^2\,dμ(x)\,\frac{dr}r \leq c\, R^n.$$ Other characterizations of uniform $n$-rectifiability in terms of smoother square functions are also obtained.
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…
We provide a sufficient geometric condition for $\mathbb{R}^n$ to be countably $(μ,m)$ rectifiable of class $\mathscr{C}^{1,α}$ (using the terminology of Federer), where $μ$ is a Radon measure having positive lower density and finite upper density $μ$ almost everywhere. Our condition involves integrals of certain many-point interaction functions (discrete curvatures) which measure flatness of simplices spanned by the parameters.
We identify a set of sufficient local conditions under which a significant portion of a Radon measure $μ$ on $\mathbb{R}^{n+1}$ with compact support can be covered by an $n$-uniformly rectifiable set at the level of a ball $B\subset \mathbb{R}^{n+1}$ such that $μ(B)\approx r(B)^n$. This result involves a flatness condition, formulated in terms of the so-called $β_1$-number of $B$, and the $L^2(μ|_B)$-boundedness, as well as a control on the mean oscillation on the ball, of the operator \begin{equation} T_μf(x)=\int \nabla_x\mathcal{E}(x,y)f(y)\,dμ(y). \end{equation} Here $\mathcal{E}(\cdot,\cd…
We study metric measure spaces that have quantitative topological control, as well as a weak form of differentiable structure. In particular, let $X$ be a pointwise doubling metric measure space. Let $U$ be a Borel subset on which the blowups of $X$ are topological planes. We show that $U$ can admit at most $2$ independent Alberti representations. Furthermore, if $U$ admits $2$ Alberti representations, then the restriction of the measure to $U$ is $2$-rectifiable. This is a partial answer to the case $n=2$ of a question of the second author and Schioppa.
In this work we show that an $n$-dimensional Borel set in Euclidean $N$-space with finite integral Menger curvature is $n$-rectifiable, meaning that it can be covered by countably many images of Lipschitz continuous functions up to a null set in the sense of Hausdorff measure. This generalises Léger's rectifiability result for one-dimensional sets to arbitrary dimension and co-dimension. In addition, we characterise possible integrands and discuss examples known from the literature. Intermediate results of independent interest include upper bounds of different versions of P. Jones's $β$-number…
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 …
Menger curvatures and $C^{1,α}$ rectifiability of measures
本文通过推进$β$数与离散曲率的关系，证明弱密度条件下逐点离散曲率的几乎处处有限性可推出测度的$C^{1,α}$ $n$-可求长性。
The purpose of this note is to record a consequence, for general metric spaces, of a recent result of David Bate. We prove the following fact: Let $X$ be a compact metric space of topological dimension $n$. Suppose that the $n$-dimensional Hausdorff measure of $X$, $\mathcal H^n(X)$, is finite. Suppose further that the lower n-density of the measure $\mathcal H^n$ is positive, $\mathcal H^n$-almost everywhere in $X$. Then $X$ contains an $n$-rectifiable subset of positive $\mathcal H^n$-measure. Moreover, the assumption on the lower density is unnecessary if one uses recently announced results…
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 s…
Characterization of rectifiability via Lusin type approximation
本文通过Lusin型逼近刻画了Radon测度的可求长性，证明其能分解为各维可求长测度当且仅当任意Lipschitz函数均可用$C^1$类函数在测度任意小误差下逼近。
Defining the $m$-th stratum of a closed subset of an $n$ dimensional Euclidean space to consist of those points, where it can be touched by a ball from at least $n-m$ linearly independent directions, we establish that the $m$-th stratum is second-order rectifiable of dimension $m$ and a Borel set. This was known for convex sets, but is new even for sets of positive reach. The result is based on a new criterion for second-order rectifiability.
Characterizing rectifiability via biLipschitz pieces of Lipschitz mappings on the space
本文给出了可求长度量空间的新刻画：通过Lipschitz映射的biLipschitz片段及完全biLipschitz分解，并利用独立Alberti表示构造方法，将不可求长空间刻画为存在与低维映射定量横截的曲线片段上$\cH^1$-零测集的空间。
Carnot rectifiability and Alberti representations
本文通过阿尔贝蒂表示与取值于卡诺群的李普希茨映射可微性，刻画了卡诺可整流性，并提出潘苏可微性空间（PDS）作为切格尔可微性空间的卡诺群类比。
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 pi…
In this work we extend many classical results concerning the relationship between densities, tangents and rectifiability to the parabolic spaces, namely $\mathbb{R}^{n+1}$ equipped with parabolic dilations. In particular we prove a Marstrand-Mattila rectifiability criterion for measures of general dimension, we provide a characterisation through densities of intrinsic rectifiable measures, and we study the structure of $1$-codimensional uniform measures. Finally, we apply some of our results to the study of a quantitative version of parabolic rectifiability: we prove that the weak constant den…
Recently, M. Badger and R. Schul proved that for a $1$-rectifiable Radon measure $μ$, the density weighted Jones' square function $$ J_{1}(x) = \mathop{\sum_{Q \in \mathcal{D}}}_{\ell(Q) \leq 1} β_{2,μ}^{2}(3Q)\frac{\ell(Q)}{μ(Q)} 1_{Q}(x) $$ is finite for $μ$-a.e. $x$. Answering a question of Badger-Schul, we show that the converse is not true. Given $ε> 0$, we construct a Radon probability measure on $[0,1]^{2} \subset \mathbb{R}^{2}$ with the properties that $J_{1}(x) \leq ε$ for all $x \in \operatorname{spt} μ$, but nevertheless the $1$-dimensional lower density of $μ$ vanishes almost ever…
Solvability of the $L^p$ Dirichlet problem for the heat equation is equivalent to parabolic uniform rectifiability in the case of a parabolic Lipschitz graph
本文证明抛物型Lipschitz图域上热量测度的$A_\infty$性质等价于边界抛物型一致可修正性，从而结合Lewis-Murray工作解决了热方程$L^p$ Dirichlet问题可解性的长期开放问题。
Interactions between quantitative rectifiability, singular integrals, and boundary value problems for harmonic functions
该论文围绕定量可求长性展开，回顾其与可求长性刻画、里斯变换L²有界性的联系及应用，还探讨了其在粗糙区域中拉普拉斯方程相关问题的L^p可解性方面的关键作用。
Parabolic uniform rectifiability is characterized by Carleson measure estimates for bounded caloric functions
本文证明抛物型均匀可修正性等价于热方程有界解的Carleson测度估计及热测度的日冕分解，在温和假设下建立了新的刻画。
Layer potentials for elliptic operators with DMO-type coefficients: big pieces $Tb$ theorem, quantitative rectifiability, and free boundary problems
本文研究了具有DMO型系数的椭圆算子的层位势，建立了测度的可求长性准则与压制$T_μ$的$Tb$定理，并以此解决了有界Wiener正则域中椭圆测度的单相及双相自由边界问题。
In this paper we show that if $μ$ is a Borel measure in $\mathbb R^{n+1}$ with growth of order $n$, so that the $n$-dimensional Riesz transform $R_μ$ is bounded in $L^2(μ)$, and $B\subset\mathbb R^{n+1}$ is a ball with $μ(B)\approx r(B)^n$ such that: (a) there is some $n$-plane $L$ passing through the center of $B$ such that for some $δ>0$ small enough, it holds $\int_B \frac{dist(x,L)}{r(B)}\,dμ(x)\leq δ\,μ(B),$ (b) for some constant $ε>0$ small enough, $\int_B |R_\mu1(x) - m_{μ,B}(R_\mu1)|^2\,dμ(x) \leq ε\,μ(B)$, where $m_{μ,B}(R_\mu1)$ stands for the mean of $R_\mu1$ on $B$ with respect to …
Square Functions and Rectifiability under Monotone Transformations of the Density
本文证明密度经区间上双Lipschitz函数复合后，原关于n-AD正则测度一致n-可求长性的平方函数Carleson条件刻画仍成立，还推广了光滑平方函数等情形的相关结论。
Weak Carleson conditions in uniformly rectifiable metric spaces: the WCD and alpha numbers
该研究通过弱Carleson条件刻画一致可求长度度量空间，证明UR空间满足弱常数密度条件，引入度量空间版alpha数并证明其弱Carleson条件可刻画度量一致可求长度性，且两结果均为更一般抽象结论的推论。
Subsets of Rectifiable curves in Hilbert Space-The Analyst's TSP
本文研究希尔伯特空间中的一维集合（豪斯多夫维数）.其目的是分类子集的希尔伯特空间所包含的一个连接集的有限豪斯多夫长度。我们这样做的扩展和改进的结果彼得琼斯和凯特Okikiolu集$\R^d$。他们的结果形成了$\R^d$定量校正的基础。我们证明了以下陈述的定量版本：有限豪斯多夫长度（或一个子集）的连通集，其特征在于以下事实：在大多数尺度上，球内的大多数点的集合，该集合位于接近直线段（这取决于球）。这是通过一个量来完成的，类似于\cite{Jones90}中引入的量，它是平方函数的几何模拟。这使我们能够得出结论，对于给定的集合$K$，这个量的$\ell_2$范数（它是$K$的函数）具有与包含$K$的最短（豪斯多夫长度）连通集相当的大小。特别是，我们的结果意味着，与正确的重新制定的定理，估计在\cite{Jones90,Okik92}是{ \bf独立的环境尺寸}。
New criteria for the rectifiability of Radon measures in terms of Riesz transforms
本文基于Riesz变换的$L^2$有界性，提出新的Radon测度可测性准则，证明在特定条件下测度可由一致可测集表示。
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