We introduce a definition of intrinsically quasi-symmetric sections in metric spaces and we prove the Ahlfors-David regularity for this class of sections. We follow a recent result by Le Donne and the author where we generalize the notion of intrinsically Lipschitz graphs in the sense of Franchi, Serapioni and Serra Cassano. We do this by focusing our attention on the graph property instead of the map one.
We investigate certain singular integral operators with Riesz-type kernels on s-dimensional Ahlfors-David regular subsets of Heisenberg groups. We show that $L^2$-boundedness, and even a little less, implies that $s$ must be an integer and the set can be approximated at some arbitrary small scales by homogeneous subgroups. It follows that the operators cannot be bounded on many self similar fractal subsets of Heisenberg groups.
Let $μ$ be an Ahlfors-David probability measure on $\mathbb{R}^q$, namely, there exist some constants $s_0>0$ and $ε_0,C_1,C_2>0$ such that \[ C_1ε^{s_0}\leqμ(B(x,ε))\leq C_2ε^{s_0},\;ε\in(0,ε_0),\;x\in{\rm supp}(μ). \] For $n\geq 1$, let $α_n$ be an $n$-optimal set for $μ$ of order $r$ and $(P_a(α_n))_{a\inα_n}$ an arbitrary Voronoi partition with respect to $α_n$. The $n$th quantization error $e_{n,r}(μ)$ for $μ$ of order $r$ is given by $e^r_{n,r}(μ):=\int d(x,α_n)^rdμ(x)$. Write \[ I_a(α,μ):=\int_{P_a(α_n)}d(x,α_n)^rdμ(x),\;a\inα_n. \] We prove that, $\underline{J}(α_n,μ):=\min_{a\inα_n}I_…
We discuss 1-Ahlfors-regular connected sets in a general metric space and prove that such sets are `flat' on most scales and in most locations. Our result is quantitative, and when combined with work of I. Hahlomaa, gives a characterization of 1-Ahlfors regular subsets of 1-Ahlfors-regular curves in metric spaces. Our result is a generalization to the metric space setting of the Analyst's (Geometric) Traveling Salesman theorems of P. Jones, K. Okikiolu, and G. David and S. Semmes, and it can be stated in terms of average Menger curvature.
We show that if an Ahlfors-David regular set $E$ of dimension $k$ has Big Pieces of Big Pieces of Lipschitz Graphs (denoted usually by $BP(BP(LG))$), then $E\subset \tilde{E}$ where $\tilde{E}$ is Ahlfors-David regular of dimension $k$ and has Big Pieces of Lipschitz Graphs (denoted usually by $BP(LG)$. Our results are quantitative and, in fact, are proven in the setting of a metric space for any family of Ahlfors-David regular sets $\mathcal{F}$ replacing $LG$. A simple corollary is the stability of the BP operator after 2 iterations. This was previously only known in the Euclidean setting fo…
Universal Ahlfors--David regularity of Steiner trees
Institute of Mathematics and Informatics, Bulgarian Academy of Sciences（保加利亚科学院数学与信息学研究所）；Saint Petersburg State University（圣彼得堡国立大学）；Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, CNRS（巴黎萨克雷大学CNRS奥赛数学实验室）
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} \le…
I prove that if $\emptyset \neq K \subset \mathbb{R}^{2}$ is a compact $s$-Ahlfors-David regular set with $s \geq 1$, then $$\dim_{\mathrm{p}} D(K) = 1,$$ where $D(K) := \{|x - y| : x,y \in K\}$ is the distance set of $K$, and $\dim_{\mathrm{p}}$ stands for packing dimension. The same proof strategy applies to other problems of similar nature. For instance, one can show that if $\emptyset \neq K \subset \mathbb{R}^{2}$ is a compact $s$-Ahlfors-David regular set with $s \geq 1$, then there exists a point $x_{0} \in K$ such that $\dim_{\mathrm{p}} K \cdot (K - x_{0}) = 1$. Specialising to produc…
This note is a contribution to large scale geometry. More precisely, we introduce the intrinsically quasi-isometric sections in metric spaces and we investigate their properties: the Ahlfors-David regularity in large scale; following Cheeger theory, it is possible to define suitable sets in order to obtain convexity and being a vector space over $\mathbb{R}$ or $\mathbb{C}$ for these sections; yet, following Cheeger's idea, we give an equivalence relation for this class of sections. Throughout the paper, we use basic mathematical tools.
We show that the Heisenberg group is not minimal in looking down. This answers Problem 11.15 in `Fractured fractals and broken dreams' by David and Semmes, or equivalently, Question 22 and hence also Question 24 in `Thirty-three yes or no questions about mappings, measures, and metrics' by Heinonen and Semmes. The non-minimality of the Heisenberg group is shown by giving an example of an Ahlfors $4$-regular metric space $X$ having big pieces of itself such that no Lipschitz map from a subset of $X$ to the Heisenberg group has image with positive measure, and by providing a Lipschitz map from t…
We introduce a notion of intrinsically Hölder graphs in metric spaces. Following a recent paper of Le Donne and the author, we prove some relevant results as the Ascoli-Arzelà compactness Theorem, Ahlfors-David regularity and the Extension Theorem for this class of sections. In the first part of this note, thanks to Cheeger theory, we define suitable sets in order to obtain a vector space over $\R$ or $\C,$ a convex set and an equivalence relation for intrinsically Hölder graphs. These last three properties are new also in the Lipschitz case. Throughout the paper, we use basic mathematical too…
Let $(X,d,μ)$ be a metric measure space. For $\emptyset\neq R\subseteq (0,\infty)$ consider the Hardy-Littlewood maximal operator $$ M_R f(x) \stackrel{\mathrm{def}}{=} \sup_{r \in R} \frac{1}{μ(B(x,r))} \int_{B(x,r)} |f| dμ.$$ We show that if there is an $n>1$ such that one has the "microdoubling condition" $ μ(B(x,(1+\frac{1}{n})r))\lesssim μ(B(x,r)) $ for all $x\in X$ and $r>0$, then the weak $(1,1)$ norm of $M_R$ has the following localization property: $$ \|M_R\|_{L_1(X) \to L_{1,\infty}(X)}\asymp \sup_{r>0} \|M_{R\cap [r,nr]}\|_{L_1(X) \to L_{1,\infty}(X)}. $$ An immediate consequence is…
The purpose of this paper is to introduce and study some basic concepts of quantitative rectifiability in the first Heisenberg group $\mathbb{H}$. In particular, we aim to demonstrate that new phenomena arise compared to the Euclidean theory, founded by G. David and S. Semmes in the 90's. The theory in $\mathbb{H}$ has an apparent connection to certain nonlinear PDEs, which do not play a role with similar questions in $\mathbb{R}^{3}$. Our main object of study are the intrinsic Lipschitz graphs in $\mathbb{H}$, introduced by B. Franchi, R. Serapioni and F. Serra Cassano in 2006. We claim that …
A well-known class of questions asks the following: If $X$ and $Y$ are metric measure spaces and $f:X\rightarrow Y$ is a Lipschitz mapping whose image has positive measure, then must $f$ have large pieces on which it is bi-Lipschitz? Building on methods of David (who is not the present author) and Semmes, we answer this question in the affirmative for Lipschitz mappings between certain types of Ahlfors $s$-regular, topological $d$-manifolds. In general, these manifolds need not be bi-Lipschitz embeddable in any Euclidean space. To prove the result, we use some facts on the Gromov-Hausdorff con…
