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 $μ$; then there exists a uniformly $n$-rectifiable subset $Γ$, with $μ(Γ\cap B)\gtrsim μ(B)$, and so that $μ|_Γ$ is absolutely continuous with respect to $H^n|_Γ$. This result is an essential tool to solve an old question on a two phase problem for harmonic measure in a subsequent paper by Azzam, Mourgoglou and Tolsa.
For 0<n<d integers and r>2, we prove that an n-dimensional Ahlfors-David regular measure M in R^d is uniformly n-rectifiable if and only if the r-variation for the Riesz transform with respect to M is a bounded operator in L^2(M). This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the L^2(M) boundedness of the Riesz transform to the uniform rectifiability of M.
We prove that if $μ$ is a d-dimensional Ahlfors-David regular measure in $\R^{d+1}$, then the boundedness of the $d$-dimensional Riesz transform in $L^2(μ)$ implies that the non-BAUP David-Semmes cells form a Carleson family. Combined with earlier results of David and Semmes, this yields the uniform rectifiability of $μ$.
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.
Let $E\subset R^d$ with $H^n(E)<\infty$, where H^n stands for the $n$-dimensional Hausdorff measure. In this paper we prove that E is n-rectifiable if and only if the limit $$\lim_{\ve\to0}\int_{y\in E:|x-y|>\ve} \frac{x-y}{|x-y|^{n+1}} dH^n(y)$$ exists H^n-almost everywhere in E. To prove this result we obtain precise estimates from above and from below for the $L^2$ norm of the n-dimensional Riesz transforms on Lipschitz graphs.
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.
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…
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 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.
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…
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 $\partialΩ$.
We consider the Calderón-Zygmund kernels $K_ {α,n}(x)=(x_i^{2n-1}/|x|^{2n-1+α})_{i=1}^d$ in $\mathbb{R}^n$ for $0<α\leq 1$ and $n\in\mathbb{N}$. We show that, on the plane, for $0<α<1$, the capacity associated to the kernels $K_{α,n}$ is comparable to the Riesz capacity $C_{\frac23(2-α),\frac 3 2}$ of non-linear potential theory. As consequences we deduce the semiadditivity and bi-Lipschitz invariance of this capacity. Furthermore we show that for any Borel set $E\subset\mathbb{R}^n$ with finite length the $L^2(\mathcal{H}^1 \lfloor E)$-boundedness of the singular integral associated to $K_{1,…
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 an Ahlfors-David regular set of dimension $n$. We show that the weak-$A_\infty$ property of harmonic measure, for the open set $Ω:= \mathbb{R}^{n+1}\setminus E$, implies uniform rectifiability of $E$. More generally, we establish a similar result for the Riesz measure, $p$-harmonic measure, associated to the $p$-Laplace operator, $1<p<\infty$.
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…
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.
For all $1\leq m\leq n-1$, we investigate the interaction of locally finite measures in $\mathbb{R}^n$ with the family of $m$-dimensional Lipschitz graphs. For instance, we characterize Radon measures $μ$, which are carried by Lipschitz graphs in the sense that there exist graphs $Γ_1,Γ_2,\dots$ such that $μ(\mathbb{R}^n\setminus\bigcup_1^\inftyΓ_i)=0$, using only countably many evaluations of the measure. This problem in geometric measure theory was classically studied within smaller classes of measures, e.g.~for the restrictions of $m$-dimensional Hausdorff measure $\mathcal{H}^m$ to $E\subs…
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 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 $E$ 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.
Let $E$ be an $1$-Ahlfors regular subset of the Heisenberg group $\mathbb{H}$. We prove that there exists a $-1$-homogeneous kernel $K_1$ such that if $E$ is contained in a $1$-regular curve the corresponding singular integral is bounded in $L^2(E)$. Conversely, we prove that there exists another $-1$-homogeneous kernel $K_2$, such that the $L^2(E)$-boundedness of its corresponding singular integral implies that $E$ is contained in an $1$-regular curve. These are the first non-Euclidean examples of kernels with such properties. Both $K_1$ and $K_2$ are weighted versions of the Riesz kernel cor…
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…
In this paper it is shown that if $μ$ is a finite Radon measure in $\mathbb R^d$ which is $n$-rectifiable and $1\leq p\leq 2$, then $$\int_0^\infty β_{μ,p}^n(x,r)^2\,\frac{dr}r<\infty \quad {for $μ$-a.e. $x\in\mathbb R^d$,}$$ where $$β_{μ,p}^n(x,r) = \inf_L (\frac1{r^n} \int_{\bar B(x,r)} (\frac{\mathrm dist(y,L)}{r})^p\,dμ(y))^{1/p},$$ with the infimum taken over all the $n$-planes $L\subset \mathbb R^d$. The $β_{μ,p}^n$ coefficients are the same as the ones considered by David and Semmes in the setting of the so called uniform $n$-rectifiability. An analogous necessary condition for $n$-rect…
In view of a recent example of a positive Radon measure $μ$ on a domain $D\subset\mathbb R^n$, $n\geqslant3$, such that $μ$ is of finite energy $E_g(μ)$ relative to the $α$-Green kernel $g$ on $D$, though the energy of $μ-μ^{D^c}$ relative to the $α$-Riesz kernel $|x-y|^{α-n}$, $0<α\leqslant2$, is not well defined (here $μ^{D^c}$ is the $α$-Riesz swept measure of $μ$ onto $D^c=\mathbb R^n\setminus D$), we propose a weaker concept of $α$-Riesz energy for which this defect has been removed. This concept is applied to the study of a minimum weak $α$-Riesz energy problem over (signed) Radon measur…
The spherical Radon-Dunkl transform $R_κ$, associated to weight functions invariant under a finite reflection group, is introduced, and some elementary properties are obtained in terms of $h$-harmonics. Several inversion formulas of $R_κ$ are given with the aid of spherical Riesz-Dunkl potentials, the Dunkl operators, and some appropriate wavelet transforms.
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…
Let R_αbe the Riesz distribution on a simple Euclidean Jordan algebra, parametrized by the complex number α. I give an elementary proof of the necessary and sufficient condition for R_αto be a locally finite complex measure (= complex Radon measure).
We study minimum energy problems relative to the $α$-Riesz kernel $|x-y|^{α-n}$, $α\in(0,2]$, over signed Radon measures $μ$ on $\mathbb R^n$, $n\geqslant3$, associated with a generalized condenser $(A_1,A_2)$, where $A_1$ is a relatively closed subset of a domain $D$ and $A_2=\mathbb R^n\setminus D$. We show that, though $A_2\cap\mathrm{Cl}_{\mathbb R^n}A_1$ may have nonzero capacity, this minimum energy problem is uniquely solvable (even in the presence of an external field) if we restrict ourselves to $μ$ with $μ^+\leqslantξ$, where a constraint $ξ$ is properly chosen. We establish the shar…
Let $μ$ be a finite Radon measure in $\mathbb{R}^d$ with polynomial growth of degree $n$, although not necessarily $n$-AD-regular. We prove that under some geometric conditions on $μ$ that are closely related to rectifiability and involve the so-called $β$-numbers of Jones, David and Semmes, all singular integral operators with an odd and sufficiently smooth Calderón-Zygmund kernel are bounded in $L^2(μ)$. As a corollary, we obtain a lower bound for the Lipschitz harmonic capacity of a compact set in $\mathbb{R}^d$ only in terms of its metric and geometric properties.
In this paper, we characterize the rectifiability (both uniform and not) of an Ahlfors regular set, E, of arbitrary co-dimension by the behavior of a regularized distance function in the complement of that set. In particular, we establish a certain version of the Riesz transform characterization of rectifiability for lower-dimensional sets. We also uncover a special situation in which the regularized distance is itself a solution to a degenerate elliptic operator in the complement of E. This allows us to precisely compute the harmonic measure of those sets associated to this degenerate operato…
For a finite collection $\mathbf A=(A_i)_{i\in I}$ of locally closed sets in $\mathbb R^n$, $n\geqslant3$, with the sign $\pm1$ prescribed such that the oppositely charged plates are mutually disjoint, we consider the minimum energy problem relative to the $α$-Riesz kernel $|x-y|^{α-n}$, $α\in(0,2]$, over positive vector Radon measures $\boldsymbolμ=(μ^i)_{i\in I}$ such that each $μ^i$, $i\in I$, is carried by $A_i$ and normalized by $μ^i(A_i)=a_i\in(0,\infty)$. We show that, though the closures of oppositely charged plates may intersect each other even in a set of nonzero capacity, this probl…
First introduced by J. Deny, the classical principle of positivity of mass states that if $κ_αμ\leqslantκ_αν$ everywhere on $\mathbb{R}^n$, then $μ(\mathbb{R}^n)\leqslantν(\mathbb{R}^n)$. Here $μ,ν$ are positive Radon measures on $\mathbb{R}^n$, $n\geqslant2$, and $κ_αμ$ is the potential of $μ$ with respect to the Riesz kernel $|x-y|^{α-n}$ of order $α\in(0,2]$, $α<n$. We strengthen Deny's principle by showing that $μ(\mathbb{R}^n)\leqslantν(\mathbb{R}^n)$ still holds even if $κ_αμ\leqslantκ_αν$ is fulfilled only on a proper subset $A$ of $\mathbb{R}^n$ that is not inner $α$-thin at infinity; …
The paper deals with minimum energy problems in the presence of external fields with respect to the Riesz kernels $|x-y|^{α-n}$, $0<α<n$, on $\mathbb R^n$, $n\geqslant2$. For quite a general (not necessarily lower semicontinuous) external field $f$, we obtain necessary and/or sufficient conditions for the existence of $λ_{A,f}$ minimizing the Gauss functional \[\int|x-y|^{α-n}\,d(μ\otimesμ)(x,y)+2\int f\,dμ\] over all positive Radon measures $μ$ with $μ(\mathbb R^n)=1$, concentrated on quite a general (not necessarily closed) $A\subset\mathbb R^n$. We also provide various alternative character…
Focusing first on the inner $α$-harmonic measure $\varepsilon_y^A$ ($\varepsilon_y$ being the unit Dirac measure, and $μ^A$ the inner $α$-Riesz balayage of a Radon measure $μ$ to $A\subset\mathbb R^n$ arbitrary), we describe its Euclidean support, provide a formula for evaluation of its total mass, establish the vague continuity of the map $y\mapsto\varepsilon_y^A$ outside the inner $α$-irregular points for $A$, and obtain necessary and sufficient conditions for $\varepsilon_y^A$ to be of finite energy (more generally, for $\varepsilon_y^A$ to be absolutely continuous with respect to inner cap…
For the Riesz kernel $κ_α(x,y):=|x-y|^{α-n}$, $0<α<n$, on $\mathbb R^n$, $n\geqslant2$, we introduce the inner pseudo-balayage $\hatω^A$ of a (Radon) measure $ω$ on $\mathbb R^n$ to a set $A\subset\mathbb R^n$ as the (unique) measure minimizing the Gauss functional \[\intκ_α(x,y)\,d(μ\otimesμ)(x,y)-2\intκ_α(x,y)\,d(ω\otimesμ)(x,y)\] over the class $\mathcal E^+(A)$ of all positive measures $μ$ of finite energy, concentrated on $A$. For quite general signed $ω$ (not necessarily of finite energy) and $A$ (not necessarily closed), such $\hatω^A$ does exist, and it maintains the basic features of …
Let $μ$ be a non-negative Radon measure on ${\mathbb R}^d$ which only satisfies the polynomial growth condition. Let ${\mathcal Y}$ be a Banach space and $H^1(μ)$ the Hardy space of Tolsa. In this paper, the authors prove that a linear operator $T$ is bounded from $H^1(μ)$ to ${\mathcal Y}$ if and only if $T$ maps all $(p, γ)$-atomic blocks into uniformly bounded elements of ${\mathcal Y}$; moreover, the authors prove that for a sublinear operator $T$ bounded from $L^1(μ)$ to $L^{1, \infty}(μ)$, if $T$ maps all $(p, γ)$-atomic blocks with $p\in(1, \infty)$ and $γ\in{\mathbb N}$ into uniformly …
We consider a uniformly elliptic operator $L_A$ in divergence form associated with an $(n+1)\times(n+1)$-matrix $A$ with real, merely bounded, and possibly non-symmetric coefficients. If $$ω_A(r)=\sup_{x\in \mathbb{R}^{n+1}} \frac{1}{|B(x,r)|}\int_{B(x,r)}\Big|A(z)-\frac{1}{|B(x,r)|}\int_{B(x,r)}A\Big|\,dz,$$ then, under suitable Dini-type assumptions on $ω_A$, we prove the following: if $μ$ is a compactly supported Radon measure in $\mathbb{R}^{n+1}$, $n \geq 2$, and $T_μf(x)=\int \nabla_xΓ_A (x,y)f(y)\, dμ(y)$ denotes the gradient of the single layer potential associated with $L_A$, then $$ …
