Naturally reductive homogeneous $(\alpha ,\beta )$-metric spaces

M. Parhizkar; H.R. Salimi Moghaddam

Displaying similar documents to “Naturally reductive homogeneous $(α, β)$ -metric spaces”

Classifying homogeneous ultrametric spaces up to coarse equivalence

Taras Banakh, Dušan Repovš (2016)

Colloquium Mathematicae

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For every metric space X we introduce two cardinal characteristics $c o v^{♭} (X)$ and $c o v^{♯} (X)$ describing the capacity of balls in X. We prove that these cardinal characteristics are invariant under coarse equivalence, and that two ultrametric spaces X,Y are coarsely equivalent if $c o v^{♭} (X) = c o v^{♯} (X) = c o v^{♭} (Y) = c o v^{♯} (Y)$ . This implies that an ultrametric space X is coarsely equivalent to an isometrically homogeneous ultrametric space if and only if $c o v^{♭} (X) = c o v^{♯} (X)$ . Moreover, two isometrically homogeneous ultrametric spaces X,Y are coarsely equivalent if and...

Some new inhomogeneous Triebel-Lizorkin spaces on metric measure spaces and their various characterizations

Dachun Yang (2005)

Studia Mathematica

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Let ${(X, ϱ, μ)}_{d, θ}$ be a space of homogeneous type, i.e. X is a set, ϱ is a quasi-metric on X with the property that there are constants θ ∈ (0,1] and C₀ > 0 such that for all x,x’,y ∈ X, $| ϱ (x, y) - ϱ (x^{'}, y) | \leq C ₀ ϱ {(x, x^{'})}^{θ} {[ϱ (x, y) + ϱ (x^{'}, y)]}^{1 - θ}$ , and μ is a nonnegative Borel regular measure on X such that for some d > 0 and all x ∈ X, $μ (y \in X : ϱ (x, y) < r) \sim r^{d}$ . Let ε ∈ (0,θ], |s| < ε and maxd/(d+ε),d/(d+s+ε) < q ≤ ∞. The author introduces new inhomogeneous Triebel-Lizorkin spaces $F_{\infty q}^{s} (X)$ and establishes their frame characterizations by first establishing a Plancherel-Pólya-type...

Commutators of Littlewood-Paley [...] g κ ∗ $g_{κ}^{*}$ -functions on non-homogeneous metric measure spaces

Guanghui Lu, Shuangping Tao (2017)

Open Mathematics

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The main purpose of this paper is to prove that the boundedness of the commutator [...] Mκ,b∗ $ℳ_{κ, b}^{*}$ generated by the Littlewood-Paley operator [...] Mκ∗ $ℳ_{κ}^{*}$ and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of [...] Mκ∗ $ℳ_{κ}^{*}$ satisfies a certain Hörmander-type condition, the authors prove that [...] Mκ,b∗ $ℳ_{κ, b}^{*}$ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from...

On the tangency of sets in generalized metric spaces for certain functions of the class $F_{ρ}^{*}$

T. Konik (1991)

Matematički Vesnik

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Type and cotype of operator spaces

Hun Hee Lee (2008)

Studia Mathematica

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We consider two operator space versions of type and cotype, namely $S_{p}$ -type, $S_{q}$ -cotype and type (p,H), cotype (q,H) for a homogeneous Hilbertian operator space H and 1 ≤ p ≤ 2 ≤ q ≤ ∞, generalizing “OH-cotype 2” of G. Pisier. We compute type and cotype of some Hilbertian operator spaces and $L_{p}$ spaces, and we investigate the relationship between a homogeneous Hilbertian space H and operator spaces with cotype (2,H). As applications we consider operator space versions of generalized little...

The discriminant and oscillation lengths for contact and Legendrian isotopies

Vincent Colin, Sheila Sandon (2015)

Journal of the European Mathematical Society

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We define an integer-valued non-degenerate bi-invariant metric (the discriminant metric) on the universal cover of the identity component of the contactomorphism group of any contact manifold. This metric has a very simple geometric definition, based on the notion of discriminant points of contactomorphisms. Using generating functions we prove that the discriminant metric is unbounded for the standard contact structures on $ℝ^{2 n} \times S^{1}$ and $ℝ P^{2 n + 1}$ . On the other hand we also show by elementary arguments...

A continuum $X$ such that $C (X)$ is not continuously homogeneous

Alejandro Illanes (2016)

Commentationes Mathematicae Universitatis Carolinae

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A metric continuum $X$ is said to be continuously homogeneous provided that for every two points $p, q \in X$ there exists a continuous surjective function $f : X \to X$ such that $f (p) = q$ . Answering a question by W.J. Charatonik and Z. Garncarek, in this paper we show a continuum $X$ such that the hyperspace of subcontinua of $X$ , $C (X)$ , is not continuously homogeneous.

Geodesically equivalent metrics on homogenous spaces

Neda Bokan, Tijana Šukilović, Srdjan Vukmirović (2019)

Czechoslovak Mathematical Journal

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Two metrics on a manifold are geodesically equivalent if the sets of their unparameterized geodesics coincide. We show that if two $G$ -invariant metrics of arbitrary signature on homogenous space $G / H$ are geodesically equivalent, they are affinely equivalent, i.e. they have the same Levi-Civita connection. We also prove that the existence of nonproportional, geodesically equivalent, $G$ -invariant metrics on homogenous space $G / H$ implies that their holonomy algebra cannot be full. We give an algorithm...

Tempered reductive homogeneous spaces

Yves Benoist, Toshiyuki Kobayashi (2015)

Journal of the European Mathematical Society

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Let $G$ be a semisimple algebraic Lie group and $H$ a reductive subgroup. We find geometrically the best even integer $p$ for which the representation of $G$ in $L^{2} (G / H)$ is almost $L^{p}$ . As an application, we give a criterion which detects whether this representation is tempered.

The nonexistence of universal metric flows

Stefan Geschke (2018)

Commentationes Mathematicae Universitatis Carolinae

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We consider dynamical systems of the form $(X, f)$ where $X$ is a compact metric space and $f : X \to X$ is either a continuous map or a homeomorphism and provide a new proof that there is no universal metric dynamical system of this kind. The same is true for metric minimal dynamical systems and for metric abstract $ω$ -limit sets, answering a question by Will Brian.

Limiting Sobolev inequalities for vector fields and canceling linear differential operators

Jean Van Schaftingen (2013)

Journal of the European Mathematical Society

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The estimate ${∥D^{k - 1} u∥}_{L^{n / (n - 1)}} \leq {∥A (D) u∥}_{L^{1}}$ is shown to hold if and only if $A (D)$ is elliptic and canceling. Here $A (D)$ is a homogeneous linear differential operator $A (D)$ of order $k$ on $ℝ^{n}$ from a vector space $V$ to a vector space $E$ . The operator $A (D)$ is defined to be canceling if $⋂_{ξ \in ℝ^{n} ∖ {0}} A (ξ) [V] = {0}$ . This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous...

Isometric embeddings of a class of separable metric spaces into Banach spaces

Sophocles K. Mercourakis, Vassiliadis G. Vassiliadis (2018)

Commentationes Mathematicae Universitatis Carolinae

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Let $(M, d)$ be a bounded countable metric space and $c > 0$ a constant, such that $d (x, y) + d (y, z) - d (x, z) \geq c$ , for any pairwise distinct points $x, y, z$ of $M$ . For such metric spaces we prove that they can be isometrically embedded into any Banach space containing an isomorphic copy of $ℓ_{\infty}$ .

Adjacent dyadic systems and the $L^{p}$ -boundedness of shift operators in metric spaces revisited

Olli Tapiola (2016)

Colloquium Mathematicae

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With the help of recent adjacent dyadic constructions by Hytönen and the author, we give an alternative proof of results of Lechner, Müller and Passenbrunner about the $L^{p}$ -boundedness of shift operators acting on functions $f \in L^{p} (X; E)$ where 1 < p < ∞, X is a metric space and E is a UMD space.

On the nontrivial solvability of systems of homogeneous linear equations over $ℤ$ in ZFC

Jan Šaroch (2020)

Commentationes Mathematicae Universitatis Carolinae

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Motivated by the paper by H. Herrlich, E. Tachtsis (2017) we investigate in ZFC the following compactness question: for which uncountable cardinals $κ$ , an arbitrary nonempty system $S$ of homogeneous $ℤ$ -linear equations is nontrivially solvable in $ℤ$ provided that each of its subsystems of cardinality less than $κ$ is nontrivially solvable in $ℤ$ ?

New Einstein metrics on $Sp (n)$ which are non-naturally reductive

Shaoxiang Zhang, Huibin Chen (2022)

Czechoslovak Mathematical Journal

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We prove that there are at least two new non-naturally reductive $Ad (Sp (l) \times Sp (k) \times Sp (k) \times Sp (k))$ invariant Einstein metrics on $Sp (l + 3 k)$ $(k < l)$ . It implies that every compact simple Lie group $Sp (n)$ for $n = l + 3 k > 4$ admits at least $2 [\frac{1}{4} (n - 1)]$ non-naturally reductive $Ad (Sp (l) \times Sp (k) \times Sp (k) \times Sp (k))$ invariant Einstein metrics.

Positively homogeneous functions and the Łojasiewicz gradient inequality

Alain Haraux (2005)

Annales Polonici Mathematici

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It is quite natural to conjecture that a positively homogeneous function with degree d ≥ 2 on $ℝ^{N}$ satisfies the Łojasiewicz gradient inequality with exponent θ = 1/d without any need for an analyticity assumption. We show that this property is true under some additional hypotheses, but not always, even for N = 2.

Metric unconditionality and Fourier analysis

Stefan Neuwirth (1998)

Studia Mathematica

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We investigate several aspects of almost 1-unconditionality. We characterize the metric unconditional approximation property (umap) in terms of “block unconditionality”. Then we focus on translation invariant subspaces $L_{E}^{p} ()$ and $C_{E} ()$ of functions on the circle and express block unconditionality as arithmetical conditions on E. Our work shows that the spaces $p_{E} ()$ , p an even integer, have a singular behaviour from the almost isometric point of view: property (umap) does not interpolate between $L_{E}^{p} ()$ ...

Some approximate fixed point theorems without continuity of the operator using auxiliary functions

Sumit Chandok, Arslan Hojjat Ansari, Tulsi Dass Narang (2019)

Mathematica Bohemica

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We introduce partial generalized convex contractions of order $4$ and rank $4$ using some auxiliary functions. We present some results on approximate fixed points and fixed points for such class of mappings having no continuity condition in $α$ -complete metric spaces and $μ$ -complete metric spaces. Also, as an application, some fixed point results in a metric space endowed with a binary relation and some approximate fixed point results in a metric space endowed with a graph have been obtained....

Displaying similar documents to “Naturally reductive homogeneous ( α , β ) -metric spaces”

Displaying similar documents to “Naturally reductive homogeneous $(α, β)$ -metric spaces”