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BMO and the John-Nirenberg Inequality on Measure Spaces Anal. Geom. Metr. Spaces (IF 0.444) Pub Date : 2020-12-31 Galia Dafni; Ryan Gibara; Andrew Lavigne
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 The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones.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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Embeddings between Triebel-Lizorkin Spaces on Metric Spaces Associated with Operators Anal. Geom. Metr. Spaces (IF 0.444) Pub Date : 2020-12-31 Athanasios G. Georgiadis; George Kyriazis
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 are
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Ultradiversification of Diversities Anal. Geom. Metr. Spaces (IF 0.444) Pub Date : 2020-12-31 Pouya Haghmaram; Kourosh Nourouzi
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. This gives
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Trace Operators on Regular Trees Anal. Geom. Metr. Spaces (IF 0.444) Pub Date : 2020-12-31 Pekka Koskela; Khanh Ngoc Nguyen; Zhuang Wang
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 0.444) Pub Date : 2020-12-31 Der-Chen Chang; Yongsheng Han; Xinfeng Wu
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 0.444) Pub Date : 2020-12-31 Ryota Kawasumi; Eiichi Nakai
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 just as
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Commutators on Weighted Morrey Spaces on Spaces of Homogeneous Type Anal. Geom. Metr. Spaces (IF 0.444) Pub Date : 2020-12-21 Ruming Gong; Ji Li; Elodie Pozzi; Manasa N. Vempati
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) with κ ∈ (0, 1) and ω ∈ Ap(X), 1 < p < ∞, if and only if b is in the BMO space. We also prove that the commutator [b, T]
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Complex Interpolation of Lizorkin-Triebel-Morrey Spaces on Domains Anal. Geom. Metr. Spaces (IF 0.444) Pub Date : 2020-11-22 Ciqiang Zhuo; Marc Hovemann; Winfried Sickel
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 0.444) Pub Date : 2020-11-09 Yoshihiro Sawano
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 counterpart
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An Intrinsic Characterization of Five Points in a CAT(0) Space Anal. Geom. Metr. Spaces (IF 0.444) Pub Date : 2020-08-27 Tetsu Toyoda
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. To prove
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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 0.444) Pub Date : 2020-08-28 Xilin Zhou; Ziyi He; Dachun Yang
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) with the optimal range p∈(ωω+η,∞) and q ∈ (0, ∞]. When and p∈(ωω+η,1]q ∈ (0, ∞], the
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Duality of Moduli and Quasiconformal Mappings in Metric Spaces Anal. Geom. Metr. Spaces (IF 0.444) Pub Date : 2020-08-24 Rebekah Jones; Panu Lahti
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 of surfaces
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Admissibility versus Ap-Conditions on Regular Trees Anal. Geom. Metr. Spaces (IF 0.444) Pub Date : 2020-08-17 Khanh Ngoc Nguyen; Zhuang Wang
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 0.444) Pub Date : 2020-08-03 Ville Suomala
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 0.444) Pub Date : 2020-07-01 Gianmarco Giovannardi
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 a
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Chordal Hausdorff Convergence and Quasihyperbolic Distance Anal. Geom. Metr. Spaces (IF 0.444) Pub Date : 2020-07-01 David A. Herron; Abigail Richard; Marie A. Snipes
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 0.444) Pub Date : 2020-03-03 Fernando Román-García
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 planes
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Integral Representation of Local Left–Invariant Functionals in Carnot Groups Anal. Geom. Metr. Spaces (IF 0.444) Pub Date : 2019-12-31 A. Maione; E. Vecchi
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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