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 converse also holds. Additionally, we present an algorithm for the construction of a rectifiable curve using appropriately chosen $δ$-nets. Throughout, we discuss how to overcome the fact that in infinite dimensional Hilbert space there may be infinitely many $δ$-separated points, even in a bounded set. Finally, we prove a characterization for pointwise doubling measures carried by Lipschitz graphs.
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…
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 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 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 …
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 $0<t<r$ satisfying $\mathscr{H}^{d}_{\infty}(E\cap B(y,t))<(2t)^{d}-\varepsilon(2r)^d$ is a Carleson set for every $\varepsilon>0$. 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}…
In this article, we address pointwise sparse domination for multilinear Calderón-Zygmund operators on upper doubling, geometrically doubling metric measure spaces. As a consequence, we have obtained sharp quantitative weighted estimates for multilinear Calderón-Zygmund operators.
In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces $\dot B^s_{p,\,q}$ and Triebel-Lizorkin spaces $\dot F^s_{p,\,q}$ for all $s\in(0,\,1)$ and $p,\,q\in(n/(n+s),\,\infty],$ both in ${\mathbb R}^n$ and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve $\dot F^s_{n/s,\,q}$ on $\rn$ for all $s\in(0,\,1)$ and $q\in(n/(n+s),\,\infty]$. A metric measure space version of the above morphism property is also established.
In this article, the authors establish the pointwise characterization of Besov and Triebel-Lizorkin spaces on spaces of homogeneous type via clarifying the relationship among Hajłasz-Sobolev spaces, Hajłasz-Besov and Hajłasz-Triebel-Lizorkin spaces, grand Besov and Triebel-Lizorkin spaces, and Besov and Triebel-Lizorkin spaces. A major novelty of this article is that all results presented in this article get rid of both the dependence on the reverse doubling condition of the measure and the metric condition of the quasi-metric under consideration. Moreover, the pointwise characterization of th…
We prove fractional Sobolev-Poincaré inequalities, capacitary versions of fractional Poincaré inequalities, and pointwise and localized fractional Hardy inequalities in a metric space equipped with a doubling measure. Our results generalize and extend earlier work where such inequalities have been considered in the Euclidean spaces or in the non-fractional setting in metric spaces. The results concerning pointwise and localized variants of fractional Hardy inequalities are new even in the Euclidean case.
On a metric measure space satisfying the doubling property, we establish several optimal characterizations of Besov and Triebel-Lizorkin spaces, including a pointwise characterization. Moreover, we discuss their (non)triviality under a Poincaré inequality.
Let ${\mathcal X}$ be a doubling metric measure space. If ${\mathcal X}$ has the $δ$-annular decay property for some $δ\in (0,\,1]$, the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, from localized spaces ${\rm BMO}_ρ({\mathcal X})$ to ${\rm BLO}_ρ({\mathcal X})$ without invoking any regularity of considered kernels. The same is true for the $g^\ast_\labda$ function and unlike the Lusin-area function, in this case, ${\mathcal X}$ is not necessary to have the $δ$-annular decay prope…
The natural maximal and minimal functions commute pointwise with the logarithm on $A_\infty$. We use this observation to characterize the spaces $A_1$ and $RH_\infty$ on metric measure spaces with a doubling measure. As the limiting cases of Muckenhoupt $A_p$ and reverse Hölder classes, respectively, their behavior is remarkably symmetric. On general metric measure spaces, an additional geometric assumption is needed in order to pass between $A_p$ and reverse Hölder descriptions. Finally, we apply the characterization to give simple proofs of several known properties of $A_1$ and $RH_\infty$, …
Given a measure $μ$ of polynomial growth, we refine a deep result by David and Mattila to construct an atomic martingale filtration of $\mathrm{supp}(μ)$ which provides the right framework for a dyadic form of nondoubling harmonic analysis. Despite this filtration being highly irregular, its atoms are comparable to balls in the given metric |which in turn are all doubling| and satisfy a weaker but crucial form of regularity. Our dyadic formulation is effective to address three basic questions: i) A dyadic form of Tolsa's RBMO space which contains it. ii) Lerner's domination and $A_2$-type boun…
We study homogeneous Besov and Triebel--Lizorkin spaces defined on doubling metric measure spaces in terms of a self-adjoint operator whose heat kernel satisfies Gaussian estimates together with its derivatives. When the measure space is a smooth manifold and such operator is a sum of squares of smooth vector fields, we prove that their intersection with $L^\infty$ is an algebra for pointwise multiplication. Our results apply to nilpotent Lie groups and Grushin settings.
