scholarly journals A Theory of Besov and Triebel-Lizorkin Spaces on Metric Measure Spaces Modeled on Carnot-Carathéodory Spaces

2008 ◽  
Vol 2008 ◽  
pp. 1-250 ◽  
Author(s):  
Yongsheng Han ◽  
Detlef Müller ◽  
Dachun Yang
Keyword(s):  
Hardy Spaces ◽  
Besov Spaces ◽  
Metric Measure Spaces ◽  
Order Zero ◽  
Wide Range ◽  
Measure Spaces ◽  
Real Method ◽  
Local Hardy Spaces ◽  
Approximation Of The Identity

We work on RD-spaces𝒳, namely, spaces of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in𝒳. An important example is the Carnot-Carathéodory space with doubling measure. By constructing an approximation of the identity with bounded support of Coifman type, we develop a theory of Besov and Triebel-Lizorkin spaces on the underlying spaces. In particular, this includes a theory of Hardy spacesHp(𝒳)and local Hardy spaceshp(𝒳)on RD-spaces, which appears to be new in this setting. Among other things, we give frame characterization of these function spaces, study interpolation of such spaces by the real method, and determine their dual spaces whenp≥1. The relations among homogeneous Besov spaces and Triebel-Lizorkin spaces, inhomogeneous Besov spaces and Triebel-Lizorkin spaces, Hardy spaces, and BMO are also presented. Moreover, we prove boundedness results on these Besov and Triebel-Lizorkin spaces for classes of singular integral operators, which include non-isotropic smoothing operators of order zero in the sense of Nagel and Stein that appear in estimates for solutions of the Kohn-Laplacian on certain classes of model domains inℂN. Our theory applies in a wide range of settings.

scholarly journals Maximal characterisation of local Hardy spaces on locally doubling manifolds

2021 ◽  
Author(s):  
Alessio Martini ◽  
Stefano Meda ◽  
Maria Vallarino
Keyword(s):  
Riemannian Manifold ◽  
Maximal Function ◽  
Hardy Spaces ◽  
Integrable Function ◽  
Local Heat ◽  
Metric Measure Spaces ◽  
Measure Spaces ◽  
Local Hardy Spaces ◽  
Atomic Hardy Space ◽  
Radial Maximal Function

AbstractWe prove a radial maximal function characterisation of the local atomic Hardy space $${{\mathfrak {h}}}^1(M)$$ h 1 ( M ) on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to $${{\mathfrak {h}}}^1(M)$$ h 1 ( M ) if and only if either its local heat maximal function or its local Poisson maximal function is integrable. A key ingredient is a decomposition of Hölder cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A. Uchiyama.

scholarly journals Commutators of Littlewood-Paley gκ∗ $g_{\kappa}^{*} $-functions on non-homogeneous metric measure spaces

2017 ◽  
Vol 15 (1) ◽  
pp. 1283-1299 ◽  
Author(s):  
Guanghui Lu ◽  
Shuangping Tao
Keyword(s):  
Hardy Spaces ◽  
Lebesgue Space ◽  
Metric Measure Spaces ◽  
Type Condition ◽  
Lebesgue Spaces ◽  
Measure Spaces ◽  
Weak Lebesgue Space ◽  
Weak Lebesgue Spaces

Abstract The main purpose of this paper is to prove that the boundedness of the commutator $\mathcal{M}_{\kappa,b}^{*} $ generated by the Littlewood-Paley operator $\mathcal{M}_{\kappa}^{*} $ 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 $\mathcal{M}_{\kappa}^{*} $ satisfies a certain Hörmander-type condition, the authors prove that $\mathcal{M}_{\kappa,b}^{*} $ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from the space L log L(μ) to the weak Lebesgue space L1,∞(μ), and is bounded from the atomic Hardy spaces H1(μ) to the weak Lebesgue spaces L1,∞(μ).

scholarly journals Parameter depending almost monotonic functions and their applications to dimensions in metric measure spaces

2009 ◽  
Vol 7 (1) ◽  
pp. 61-89 ◽  
Author(s):  
Natasha Samko
Keyword(s):  
Operator Theory ◽  
Sufficient Conditions ◽  
Metric Measure Spaces ◽  
Lower And Upper Bounds ◽  
Index Numbers ◽  
Weights And Measures ◽  
Monotonic Functions ◽  
Measure Spaces ◽  
Necessary And Sufficient

In connection with application to various problems of operator theory, we study almost monotonic functionsw(x, r) depending on a parameterxwhich runs a metric measure spaceX, and the so called index numbersm(w, x),M(w, x) of such functions, and consider some generalized Zygmund, Bary, Lozinskii and Stechkin conditions. The main results contain necessary and sufficient conditions, in terms of lower and upper bounds of indicesm(w, x) andM(w, x) , for the uniform belongness of functionsw(·,r) to Zygmund-Bary-Stechkin classes. We give also applications to local dimensions in metric measure spaces and characterization of some integral inequalities involving radial weights and measures of balls in such spaces.

The distance of L∞ from BMO on metric measure spaces

2014 ◽  
Vol 5 (2) ◽  
Author(s):  
Juha Kinnunen ◽  
Parantap Shukla
Keyword(s):  
Metric Measure Spaces ◽  
Measure Spaces

Abstract.This work has two principal aims. The first is to obtain a characterization of the closure of

scholarly journals SMOOTH METRIC MEASURE SPACES AND QUASI-EINSTEIN METRICS

2012 ◽  
Vol 23 (10) ◽  
pp. 1250110 ◽  
Author(s):  
JEFFREY S. CASE
Keyword(s):  
Einstein Metrics ◽  
Conformal Geometry ◽  
Point Of View ◽  
Metric Measure Spaces ◽  
Unified Framework ◽  
Variational Characterization ◽  
Gradient Ricci Solitons ◽  
Smooth Metric Measure Spaces ◽  
Measure Spaces

Smooth metric measure spaces have been studied from the two different perspectives of Bakry–Émery and Chang–Gursky–Yang, both of which are closely related to work of Perelman on the Ricci flow. These perspectives include a generalization of the Ricci curvature and the associated quasi-Einstein metrics, which include Einstein metrics, conformally Einstein metrics, gradient Ricci solitons and static metrics. In this paper, we describe a natural perspective on smooth metric measure spaces from the point of view of conformal geometry and show how it unites these earlier perspectives within a unified framework. We offer many results and interpretations which illustrate the unifying nature of this perspective, including a natural variational characterization of quasi-Einstein metrics as well as some interesting families of examples of such metrics.

scholarly journals Boundedness of Commutators of Marcinkiewicz Integrals on Nonhomogeneous Metric Measure Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-12 ◽  
Author(s):  
Guanghui Lu ◽  
Shuangping Tao
Keyword(s):  
Hardy Spaces ◽  
Measure Space ◽  
Morrey Spaces ◽  
Marcinkiewicz Integral ◽  
Metric Measure Spaces ◽  
Doubling Condition ◽  
Lebesgue Spaces ◽  
Marcinkiewicz Integrals ◽  
Measure Spaces ◽  
Weak Lebesgue Spaces

Let(X,d,μ)be a metric measure space satisfying the upper doubling condition and geometrically doubling condition in the sense of Hytönen. The aim of this paper is to establish the boundedness of commutatorMbgenerated by the Marcinkiewicz integralMand Lipschitz functionb. The authors prove thatMbis bounded from the Lebesgue spacesLp(μ)to weak Lebesgue spacesLq(μ)for1≤p<n/β, from the Lebesgue spacesLp(μ)to the spacesRBMO(μ)forp=n/β, and from the Lebesgue spacesLp(μ)to the Lipschitz spacesLip(β-n/p)(μ)forn/β<p≤∞. Moreover, some results in Morrey spaces and Hardy spaces are also discussed.

Some Notes on Function Spaces and Dirichlet Forms on Self-Similar Sets

2006 ◽  
Vol 6 (3) ◽  
Author(s):  
Yuan Liu ◽  
Zhi-Ying Wen
Keyword(s):  
Dirichlet Forms ◽  
Function Spaces ◽  
Metric Measure Spaces ◽  
Kernel Estimates ◽  
Self Similarity ◽  
Equivalent Characterization ◽  
Measure Spaces ◽  
Self Similar ◽  
Sierpinski Gaskets

AbstractGrigor’yan, Hu and Lau [10] introduced sub-Gaussian heat kernels on general metric measure spaces and defined a family of function spaces to characterize the domain of associated Dirichlet forms. In this paper, we will improve their results about norm equivalence. As an application, we construct self-similar Dirichlet forms on a class of self-similar sets containing the Sierpiński gaskets and carpets. Then we prove the Poincaré inequality and give effective resistance estimates by the self-similarity. Consequently, we have a new equivalent characterization of heat kernel estimates through function spaces with strong recurrent condition.

scholarly journals Characterization of Low Dimensional RCD*(K, N) Spaces

2016 ◽  
Vol 4 (1) ◽  
Author(s):  
Yu Kitabeppu ◽  
Sajjad Lakzian
Keyword(s):  
Hausdorff Dimension ◽  
Isoperimetric Inequality ◽  
Type Inequality ◽  
Metric Measure Spaces ◽  
One Dimensional ◽  
Curvature Bounds ◽  
Measure Spaces ◽  
Low Dimensional ◽  
Regular Sets

AbstractIn this paper,we give the characterization of metric measure spaces that satisfy synthetic lower Riemannian Ricci curvature bounds (so called RCD*(K, N) spaces) with non-empty one dimensional regular sets. In particular, we prove that the class of Ricci limit spaces with Ric ≥ K and Hausdorff dimension N and the class of RCD*(K, N) spaces coincide for N < 2 (They can be either complete intervals or circles). We will also prove a Bishop-Gromov type inequality (that is ,roughly speaking, a converse to the Lévy-Gromov’s isoperimetric inequality and was previously only known for Ricci limit spaces) which might be also of independent interest.

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