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C 1,α-rectifiability in low codimension in Heisenberg groups Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2024-02-22 Kennedy Obinna Idu, Francesco Paolo Maiale
A natural higher-order notion of C 1 , α {C}^{1,\alpha } -rectifiability, 0 < α ≤ 1 0\lt \alpha \le 1 , is introduced for subsets of the Heisenberg groups H n {{\mathbb{H}}}^{n} in terms of covering a set almost everywhere with a countable union of ( C H 1 , α , H ) \left({{\bf{C}}}_{H}^{1,\alpha },{\mathbb{H}}) -regular surfaces. Using this, we prove a geometric characterization of C 1 , α {C}^{1
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Extremal subsets in geodesically complete spaces with curvature bounded above Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2024-02-15 Tadashi Fujioka
We introduce the notion of an extremal subset in a geodesically complete space with curvature bounded above, i.e., a GCBA space. This is an analog of an extremal subset in an Alexandrov space with curvature bounded below introduced by Perelman and Petrunin. We prove that under an additional assumption, the set of topological singularities in a GCBA space forms an extremal subset. We also exhibit some
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Pseudometric spaces: From minimality to maximality in the groups of combinatorial self-similarities Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2024-02-10 Viktoriia Bilet, Oleksiy Dovgoshey
The group of combinatorial self-similarities of a pseudometric space ( X , d ) \left(X,d) is the maximal subgroup of the symmetric group Sym ( X ) {\rm{Sym}}\left(X) whose elements preserve the four-point equality d ( x , y ) = d ( u , v ) d\left(x,y)=d\left(u,v) . Let us denote by ℐP {\mathcal{ {\mathcal I} P}} the class of all pseudometric spaces ( X , d ) \left(X,d) for which every combinatorial
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Estimates for bilinear θ-type generalized fractional integral and its commutator on new non-homogeneous generalized Morrey spaces Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2023-11-24 Guanghui Lu, Miaomiao Wang, Shuangping Tao
Let ( X , d , μ ) \left({\mathcal{X}},d,\mu ) be a non-homogeneous metric measure space satisfying the geometrically doubling and upper doubling conditions. In this setting, we first introduce a generalized Morrey space M p u ( μ ) {M}_{p}^{u}\left(\mu ) , where 1 ≤ p < ∞ 1\le p\lt \infty and u ( x , r ) : X × ( 0 , ∞ ) → ( 0 , ∞ ) u\left(x,r):{\mathcal{X}}\times \left(0,\infty )\to \left(0,\infty
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Identifying 1-rectifiable measures in Carnot groups Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2023-11-22 Matthew Badger, Sean Li, Scott Zimmerman
We continue to develop a program in geometric measure theory that seeks to identify how measures in a space interact with canonical families of sets in the space. In particular, extending a theorem of M. Badger and R. Schul in Euclidean space, for an arbitrary locally finite Borel measure in an arbitrary Carnot group, we develop tests that identify the part of the measure that is carried by rectifiable
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Characterization of Lipschitz functions via the commutators of multilinear fractional integral operators in variable Lebesgue spaces Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2023-07-05 Pu Zhang, Jianglong Wu
The main purpose of this article is to establish some new characterizations of the (variable) Lipschitz spaces in terms of the boundedness of commutator of multilinear fractional Calderón-Zygmund integral operators in the context of the variable exponent Lebesgue spaces. The authors do so by applying the techniques of Fourier series and multilinear fractional integral operator, as well as some pointwise
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Exceptional families of measures on Carnot groups Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2023-05-25 Bruno Franchi, Irina Markina
We study the families of measures on Carnot groups that have vanishing p p -module, which we call M p {M}_{p} -exceptional families. We found necessary and sufficient Conditions for the family of intrinsic Lipschitz surfaces passing through a common point to be M p {M}_{p} -exceptional for p ≥ 1 p\ge 1 . We describe a wide class of M p {M}_{p} -exceptional intrinsic Lipschitz surfaces for p ∈ ( 0
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Convergence theorems for monotone vector field inclusions and minimization problems in Hadamard spaces Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2023-04-28 Sani Salisu, Poom Kumam, Songpon Sriwongsa
This article analyses two schemes: Mann-type and viscosity-type proximal point algorithms. Using these schemes, we establish Δ-convergence and strong convergence theorems for finding a common solution of monotone vector field inclusion problems, a minimization problem, and a common fixed point of multivalued demicontractive mappings in Hadamard spaces. We apply our results to find mean and median values
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On the problem of prescribing weighted scalar curvature and the weighted Yamabe flow Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2023-04-28 Pak Tung Ho, Jinwoo Shin
The weighted Yamabe problem introduced by Case is the generalization of the Gagliardo-Nirenberg inequalities to smooth metric measure spaces. More precisely, given a smooth metric measure space ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi }{\rm{d}}{V}_{g},m) , the weighted Yamabe problem consists on finding another smooth metric measure space conformal to ( M , g , e − ϕ d V g , m ) \left(M,g,{e}^{-\phi
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A non-geodesic analogue of Reshetnyak’s majorization theorem Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2023-03-28 Tetsu Toyoda
For any real number κ \kappa and any integer n ≥ 4 n\ge 4 , the Cycl n ( κ ) {{\rm{Cycl}}}_{n}\left(\kappa ) condition introduced by Gromov (CAT(κ)-spaces: construction and concentration, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 280 (2001), (Geom. i Topol. 7), 100–140, 299–300) is a necessary condition for a metric space to admit an isometric embedding into a CAT ( κ ) {\rm{CAT}}\left(\kappa
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Separation functions and mild topologies Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2023-03-22 Andrea C. G. Mennucci
Given M M and N N Hausdorff topological spaces, we study topologies on the space C 0 ( M ; N ) {C}^{0}\left(M;\hspace{0.33em}N) of continuous maps f : M → N f:M\to N . We review two classical topologies, the “strong” and the “weak” topology. We propose a definition of “mild topology” that is coarser than the “strong” and finer than the “weak” topology. We compare properties of these three topologies
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Potential Theory on Gromov Hyperbolic Spaces Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-11-28 Matthias Kemper, Joachim Lohkamp
Gromov hyperbolic spaces have become an essential concept in geometry, topology and group theory. Herewe extend Ancona’s potential theory on Gromov hyperbolic manifolds and graphs of bounded geometry to a large class of Schrödinger operators on Gromov hyperbolic metric measure spaces, unifying these settings in a common framework ready for applications to singular spaces such as RCD spaces or minimal
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Asymptotically Mean Value Harmonic Functions in Doubling Metric Measure Spaces Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-11-08 Tomasz Adamowicz, Antoni Kijowski, Elefterios Soultanis
We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajłasz–Sobolev
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Isoperimetric and Poincaré Inequalities on Non-Self-Similar Sierpiński Sponges: the Borderline Case Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-11-08 Sylvester Eriksson-Bique, Jasun Gong
In this paper we construct a large family of examples of subsets of Euclidean space that support a 1-Poincaré inequality yet have empty interior. These examples are formed from an iterative process that involves removing well-behaved domains, or more precisely, domains whose complements are uniform in the sense of Martio and Sarvas. While existing arguments rely on explicit constructions of Semmes
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Inverse Gauss Curvature Flows and Orlicz Minkowski Problem Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-11-08 Bin Chen, Jingshi Cui, Peibiao Zhao
Liu and Lu [27] investigated a generalized Gauss curvature flow and obtained an even solution to the dual Orlicz-Minkowski problem under some appropriate assumptions. The present paper investigates a inverse Gauss curvature flow, and achieves the long-time existence and convergence of this flow via a different C 0-estimate technique under weaker conditions. As an application of this inverse Gauss curvature
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On L 1-Embeddability of Unions of L 1-Embeddable Metric Spaces and of Twisted Unions of Hypercubes Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-10-11 Mikhail I. Ostrovskii, Beata Randrianantoanina
We study properties of twisted unions of metric spaces introduced in [Johnson, Lindenstrauss, and Schechtman 1986], and in [Naor and Rabani 2017]. In particular, we prove that under certain natural mild assumptions twisted unions of L 1-embeddable metric spaces also embed in L 1 with distortions bounded above by constants that do not depend on the metric spaces themselves, or on their size, but only
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Certain Conditions for a Finsler Manifold to Be Isometric with a Finsler Sphere Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-09-30 Songting Yin, Huarong Wang
We show that if there is a smooth function f on a Finsler n -space M satisfying Δ2 f = − kfg Δ f for a positive constant k , then M is diffeomorphic with the n -sphere 𝕊 n , where g denotes the weighted Riemannian metric. Moreover, we further show that the manifold is isometric to a Finsler sphere if the Ricci curvature is bounded below by ( n − [one.tf]) k and the S -curvature vanishes.
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Lipschitz Chain Approximation of Metric Integral Currents Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-09-30 Tommaso Goldhirsch
Every integral current in a locally compact metric space X can be approximated by a Lipschitz chain with respect to the normal mass, provided that Lipschitz maps into X can be extended slightly.
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A Cornucopia of Carnot Groups in Low Dimensions Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-09-30 Enrico Le Donne, Francesca Tripaldi
Stratified groups are those simply connected Lie groups whose Lie algebras admit a derivation for which the eigenspace with eigenvalue 1 is Lie generating. When a stratified group is equipped with a left-invariant path distance that is homogeneous with respect to the automorphisms induced by the derivation, this metric space is known as Carnot group. Carnot groups appear in several mathematical contexts
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Conformal Transformation of Uniform Domains Under Weights That Depend on Distance to The Boundary Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-09-30 Ryan Gibara, Nageswari Shanmugalingam
The sphericalization procedure converts a Euclidean space into a compact sphere. In this note we propose a variant of this procedure for locally compact, rectifiably path-connected, non-complete, unbounded metric spaces by using conformal deformations that depend only on the distance to the boundary of the metric space. This deformation is locally bi-Lipschitz to the original domain near its boundary
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Growth Competitions on Spherically Symmetric Riemannian Manifolds Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-09-19 Rotem Assouline
We propose a model for a growth competition between two subsets of a Riemannian manifold. The sets grow at two different rates, avoiding each other. It is shown that if the competition takes place on a surface which is rotationally symmetric about the starting point of the slower set, then if the surface is conformally equivalent to the Euclidean plane, the slower set remains in a bounded region, while
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Fractional Type Marcinkiewicz Integral Operator Associated with Θ-Type Generalized Fractional Kernel and Its Commutator on Non-homogeneous Spaces Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-08-15 Guanghui Lu, Shuangping Tao, Miaomiao Wang
Let (𝒳, d, μ) be a non-homogeneous metric measure space satisfying the upper doubling and geometrically doubling conditions in the sense of Hytönen. Under assumption that θ and dominating function λ satisfy certain conditions, the authors prove that fractional type Marcinkiewicz integral operator M ˜ \tilde M α,lρ,q associated with θ-type generalized fractional kernel is bounded from the generalized
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Branching Geodesics of the Gromov-Hausdorff Distance Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-08-08 Yoshito Ishiki
In this paper, we first evaluate topological distributions of the sets of all doubling spaces, all uniformly disconnected spaces, and all uniformly perfect spaces in the space of all isometry classes of compact metric spaces equipped with the Gromov–Hausdorff distance.We then construct branching geodesics of the Gromov–Hausdorff distance continuously parameterized by the Hilbert cube, passing through
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Properties of Functions on a Bounded Charge Space Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-06-10 Jonathan M. Keith
A charge space (X, 𝒜, µ) is a generalisation of a measure space, consisting of a sample space X, a field of subsets 𝒜 and a finitely additive measure µ, also known as a charge. Properties a real-valued function on X may possess include T 1-measurability and integrability. However, these properties are less well studied than their measure-theoretic counterparts. This paper describes new characterisations
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A Simple Proof of Dvoretzky-Type Theorem for Hausdorff Dimension in Doubling Spaces Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-04-26 Manor Mendel
The ultrametric skeleton theorem [Mendel, Naor 2013] implies, among other things, the following nonlinear Dvoretzky-type theorem for Hausdorff dimension: For any 0 < β < α, any compact metric space X of Hausdorff dimension α contains a subset which is biLipschitz equivalent to an ultrametric and has Hausdorff dimension at least β. In this note we present a simple proof of the ultrametric skeleton theorem
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Non-Parametric Mean Curvature Flow with Prescribed Contact Angle in Riemannian Products Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-03-08 Jean-Baptiste Casteras, Esko Heinonen, Ilkka Holopainen, Jorge H. De Lira
Assuming that there exists a translating soliton u ∞ with speed C in a domain Ω and with prescribed contact angle on ∂Ω, we prove that a graphical solution to the mean curvature flow with the same prescribed contact angle converges to u ∞ + Ct as t →∞. We also generalize the recent existence result of Gao, Ma, Wang and Weng to non-Euclidean settings under suitable bounds on convexity of Ω and Ricci
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Comparison Theorems on Weighted Finsler Manifolds and Spacetimes with ϵ-Range Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2022-03-05 Yufeng Lu, Ettore Minguzzi, Shin-ichi Ohta
We establish the Bonnet–Myers theorem, Laplacian comparison theorem, and Bishop–Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of weighted Ricci curvature bounded below by using the weight function. These comparison theorems are formulated with ϵ-range introduced in our previous paper, that provides a natural viewpoint of interpolating weighted
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Quasiconformal Jordan Domains Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2021-01-01 Toni Ikonen
We extend the classical Carathéodory extension theorem to quasiconformal Jordan domains ( Y , d Y ). We say that a metric space ( Y , d Y ) is a quasiconformal Jordan domain if the completion ̄ Y of ( Y , d Y ) has finite Hausdorff 2-measure, the boundary ∂ Y = ̄ Y \ Y is homeomorphic to 𝕊 1 , and there exists a homeomorphism ϕ : 𝔻 →( Y , d Y ) that is quasiconformal in the geometric sense. We show
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Variable Anisotropic Hardy Spaces with Variable Exponents Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2021-01-01 Zhenzhen Yang, Yajuan Yang, Jiawei Sun, Baode Li
Let p (·) : ℝ n → (0, ∞] be a variable exponent function satisfying the globally log-Hölder continuous and let Θ be a continuous multi-level ellipsoid cover of ℝ n introduced by Dekel et al. [12]. In this article, we introduce highly geometric Hardy spaces H p (·) ( Θ ) via the radial grand maximal function and then obtain its atomic decomposition, which generalizes that of Hardy spaces H p ( Θ ) on
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Hölder Parameterization of Iterated Function Systems and a Self-Affine Phenomenon Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2021-01-01 Matthew Badger, Vyron Vellis
We investigate the Hölder geometry of curves generated by iterated function systems (IFS) in a complete metric space. A theorem of Hata from 1985 asserts that every connected attractor of an IFS is locally connected and path-connected. We give a quantitative strengthening of Hata’s theorem. First we prove that every connected attractor of an IFS is (1/ s )-Hölder path-connected, where s is the similarity
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On Weak Super Ricci Flow through Neckpinch Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2021-01-01 Sajjad Lakzian, Michael Munn
In this article, we study the Ricci flow neckpinch in the context of metric measure spaces. We introduce the notion of a Ricci flow metric measure spacetime and of a weak (refined) super Ricci flow associated to convex cost functions (cost functions which are increasing convex functions of the distance function). Our definition of a weak super Ricci flow is based on the coupled contraction property
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5-Point CAT(0) Spaces after Tetsu Toyoda Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2021-01-01 Nina Lebedeva, Anton Petrunin
We give another proof of Toyoda’s theorem that describes 5-point subspaces in CAT(0) length spaces.
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Concentration of Product Spaces Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2021-01-01 Daisuke Kazukawa
We investigate the relation between the concentration and the product of metric measure spaces. We have the natural question whether, for two concentrating sequences of metric measure spaces, the sequence of their product spaces also concentrates. A partial answer is mentioned in Gromov’s book [4]. We obtain a complete answer for this question.
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Dilation Type Inequalities for Strongly-Convex Sets in Weighted Riemannian Manifolds Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2021-01-01 Hiroshi Tsuji
In this paper, we consider a dilation type inequality on a weighted Riemannian manifold, which is classically known as Borell’s lemma in high-dimensional convex geometry. We investigate the dilation type inequality as an isoperimetric type inequality by introducing the dilation profile and estimate it by the one for the corresponding model space under lower weighted Ricci curvature bounds. We also
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Density and Extension of Differentiable Functions on Metric Measure Spaces Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2021-01-01 Rafael Espínola García, Luis Sánchez González
We consider vector valued mappings defined on metric measure spaces with a measurable differentiable structure and study both approximations by nicer mappings and regular extensions of the given mappings when defined on closed subsets. Therefore, we propose a first approach to these problems, largely studied on Euclidean and Banach spaces during the last century, for first order differentiable functions
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Remarks on Manifolds with Two-Sided Curvature Bounds Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2021-01-01 Vitali Kapovitch, Alexander Lytchak
We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.
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Sub-Finsler Horofunction Boundaries of the Heisenberg Group Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2021-01-01 Nate Fisher, Sebastiano Nicolussi Golo
We give a complete analytic and geometric description of the horofunction boundary for polygonal sub-Finsler metrics, that is, those that arise as asymptotic cones of word metrics, on the Heisenberg group. We develop theory for the more general case of horofunction boundaries in homogeneous groups by connecting horofunctions to Pansu derivatives of the distance function.
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On the Volume of Sections of the Cube Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2021-01-01 Grigory Ivanov, Igor Tsiutsiurupa
We study the properties of the maximal volume k -dimensional sections of the n -dimensional cube [−1, 1] n . We obtain a first order necessary condition for a k -dimensional subspace to be a local maximizer of the volume of such sections, which we formulate in a geometric way. We estimate the length of the projection of a vector of the standard basis of ℝ n onto a k -dimensional subspace that maximizes
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Embeddings between Triebel-Lizorkin Spaces on Metric Spaces Associated with Operators Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Athanasios G. Georgiadis, George Kyriazis
Abstract We consider the general framework of a metric measure space satisfying the doubling volume property, associated with a non-negative self-adjoint operator, whose heat kernel enjoys standard Gaussian localization. We prove embedding theorems between Triebel-Lizorkin spaces associated with operators. Embeddings for non-classical Triebel-Lizorkin and (both classical and non-classical) Besov spaces
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Ultradiversification of Diversities Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Pouya Haghmaram, Kourosh Nourouzi
Abstract In this paper, using the idea of ultrametrization of metric spaces we introduce ultradiversification of diversities. We show that every diversity has an ultradiversification which is the greatest nonexpansive ultra-diversity image of it. We also investigate a Hausdorff-Bayod type problem in the setting of diversities, namely, determining what diversities admit a subdominant ultradiversity
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Trace Operators on Regular Trees Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Pekka Koskela, Khanh Ngoc Nguyen, Zhuang Wang
Abstract We consider different notions of boundary traces for functions in Sobolev spaces defined on regular trees and show that the almost everywhere existence of these traces is independent of the chosen definition of a trace.
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Construction of Frames on the Heisenberg Groups Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Der-Chen Chang, Yongsheng Han, Xinfeng Wu
Abstract In this paper, we present a construction of frames on the Heisenberg group without using the Fourier transform. Our methods are based on the Calderón-Zygmund operator theory and Coifman’s decomposition of the identity operator on the Heisenberg group. These methods are expected to be used in further studies of several complex variables.
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Pointwise Multipliers on Weak Morrey Spaces Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Ryota Kawasumi, Eiichi Nakai
Abstract We consider generalized weak Morrey spaces with variable growth condition on spaces of homogeneous type and characterize the pointwise multipliers from a generalized weak Morrey space to another one. The set of all pointwise multipliers from a weak Lebesgue space to another one is also a weak Lebesgue space. However, we point out that the weak Morrey spaces do not always have this property
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BMO and the John-Nirenberg Inequality on Measure Spaces Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Galia Dafni, Ryan Gibara, Andrew Lavigne
Abstract We study the space BMO𝒢 (𝕏) in the general setting of a measure space 𝕏 with a fixed collection 𝒢 of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in 𝒢. The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones. In particular, three aspects of BMO are considered:
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Commutators on Weighted Morrey Spaces on Spaces of Homogeneous Type Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Ruming Gong, Ji Li, Elodie Pozzi, Manasa N. Vempati
Abstract In this paper, we study the boundedness and compactness of the commutator of Calderón– Zygmund operators T on spaces of homogeneous type (X, d, µ) in the sense of Coifman and Weiss. More precisely, we show that the commutator [b, T] is bounded on the weighted Morrey space Lωp,k(X) L_\omega ^{p,k}\left( X \right) with κ ∈ (0, 1) and ω ∈ Ap(X), 1 < p < ∞, if and only if b is in the BMO space
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Complex Interpolation of Lizorkin-Triebel-Morrey Spaces on Domains Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Ciqiang Zhuo, Marc Hovemann, Winfried Sickel
Abstract In this article the authors study complex interpolation of Sobolev-Morrey spaces and their generalizations, Lizorkin-Triebel-Morrey spaces. Both scales are considered on bounded domains. Under certain conditions on the parameters the outcome belongs to the scale of the so-called diamond spaces.
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A Weak Type Vector-Valued Inequality for the Modified Hardy–Littlewood Maximal Operator for General Radon Measure on ℝn Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Yoshihiro Sawano
Abstract The aim of this paper is to prove the weak type vector-valued inequality for the modified Hardy– Littlewood maximal operator for general Radon measure on ℝn. Earlier, the strong type vector-valued inequality for the same operator and the weak type vector-valued inequality for the dyadic maximal operator were obtained. This paper will supplement these existing results by proving a weak type
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Real-Variable Characterizations of Hardy–Lorentz Spaces on Spaces of Homogeneous Type with Applications to Real Interpolation and Boundedness of Calderón–Zygmund Operators Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Xilin Zhou, Ziyi He, Dachun Yang
Abstract Let (𝒳, d, μ) be a space of homogeneous type, in the sense of Coifman and Weiss, with the upper dimension ω. Assume that η ∈(0, 1) is the smoothness index of the wavelets on 𝒳 constructed by Auscher and Hytönen. In this article, via grand maximal functions, the authors introduce the Hardy–Lorentz spaces H*p,q(𝒳) H_*^{p,q}\left( \mathcal{X} \right) with the optimal range p∈(ωω+η,∞) p \in
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An Intrinsic Characterization of Five Points in a CAT(0) Space Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Tetsu Toyoda
Abstract Gromov (2001) and Sturm (2003) proved that any four points in a CAT(0) space satisfy a certain family of inequalities. We call those inequalities the ⊠-inequalities, following the notation used by Gromov. In this paper, we prove that a metric space X containing at most five points admits an isometric embedding into a CAT(0) space if and only if any four points in X satisfy the ⊠-inequalities
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Duality of Moduli and Quasiconformal Mappings in Metric Spaces Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Rebekah Jones, Panu Lahti
Abstract We prove a duality relation for the moduli of the family of curves connecting two sets and the family of surfaces separating the sets, in the setting of a complete metric space equipped with a doubling measure and supporting a Poincaré inequality. Then we apply this to show that quasiconformal mappings can be characterized by the fact that they quasi-preserve the modulus of certain families
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Admissibility versus Ap-Conditions on Regular Trees Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Khanh Ngoc Nguyen, Zhuang Wang
Abstract We show that the combination of doubling and (1, p)-Poincaré inequality is equivalent to a version of the Ap-condition on rooted K-ary trees.
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Intermediate Value Property for the Assouad Dimension of Measures Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Ville Suomala
Abstract Hare, Mendivil, and Zuberman have recently shown that if X ⊂ ℝ is compact and of non-zero Assouad dimension dimA X, then for all s > dimA X, X supports measures with Assouad dimension s. We generalize this result to arbitrary complete metric spaces.
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Higher Dimensional Holonomy Map for Rules Submanifolds in Graded Manifolds Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Gianmarco Giovannardi
Abstract The deformability condition for submanifolds of fixed degree immersed in a graded manifold can be expressed as a system of first order PDEs. In the particular but important case of ruled submanifolds, we introduce a natural choice of coordinates, which allows to deeply simplify the formal expression of the system, and to reduce it to a system of ODEs along a characteristic direction. We introduce
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Chordal Hausdorff Convergence and Quasihyperbolic Distance Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 David A. Herron, Abigail Richard, Marie A. Snipes
Abstract We study Hausdorff convergence (and related topics) in the chordalization of a metric space to better understand pointed Gromov-Hausdorff convergence of quasihyperbolic distances (and other conformal distances).
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Intersections of Projections and Slicing Theorems for the Isotropic Grassmannian and the Heisenberg group Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2020-01-01 Fernando Román-García
Abstract This paper studies the Hausdorff dimension of the intersection of isotropic projections of subsets of ℝ2n, as well as dimension of intersections of sets with isotropic planes. It is shown that if A and B are Borel subsets of ℝ2n of dimension greater than m, then for a positive measure set of isotropic m-planes, the intersection of the images of A and B under orthogonal projections onto these
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Integral Representation of Local Left–Invariant Functionals in Carnot Groups Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2019-12-31 A. Maione, E. Vecchi
Abstract The aim of this note is to prove a representation theorem for left–invariant functionals in Carnot groups. As a direct consequence, we can also provide a Г-convergence result for a smaller class of functionals.
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Long-Scale Ollivier Ricci Curvature of Graphs Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2019-03-01 D. Cushing, S. Kamtue
Abstract We study the long-scale Ollivier Ricci curvature of graphs as a function of the chosen idleness. Similarly to the previous work on the short-scale case, we show that this idleness function is concave and piecewise linear with at most 3 linear parts. We provide bounds on the length of the first and last linear pieces. We also study the long-scale curvature for the Cartesian product of two regular
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Geometry of Generated Groups with Metrics Induced by Their Cayley Color Graphs Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2019-03-01 Teerapong Suksumran
Abstract Let G be a group and let S be a generating set of G. In this article,we introduce a metric dC on G with respect to S, called the cardinal metric.We then compare geometric structures of (G, dC) and (G, dW), where dW denotes the word metric. In particular, we prove that if S is finite, then (G, dC) and (G, dW) are not quasiisometric in the case when (G, dW) has infinite diameter and they are
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Distance Bounds for Graphs with Some Negative Bakry-Émery Curvature Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2019-03-01 Shiping Liu, Florentin Münch, Norbert Peyerimhoff, Christian Rose
Abstract We prove distance bounds for graphs possessing positive Bakry-Émery curvature apart from an exceptional set, where the curvature is allowed to be non-positive. If the set of non-positively curved vertices is finite, then the graph admits an explicit upper bound for the diameter. Otherwise, the graph is a subset of the tubular neighborhood with an explicit radius around the non-positively curved
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Group Approximation in Cayley Topology and Coarse Geometry, Part II: Fibred Coarse Embeddings Anal. Geom. Metr. Spaces (IF 1.0) Pub Date : 2019-01-01 Masato Mimura, Hiroki Sako
Abstract The objective of this series is to study metric geometric properties of disjoint unions of Cayley graphs of amenable groups by group properties of the Cayley accumulation points in the space of marked groups. In this Part II, we prove that a disjoint union admits a fibred coarse embedding into a Hilbert space (as a disjoint union) if and only if the Cayley boundary of the sequence in the space