scholarly journals Local Morrey and Campanato Spaces on Quasimetric Measure Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-15
Author(s):  
Krzysztof Stempak ◽  
Xiangxing Tao
Keyword(s):  
Structural Properties ◽  
Measure Space ◽  
Morrey Spaces ◽  
Maximal Operators ◽  
Measure Spaces ◽  
Local Morrey Spaces

We define and investigate generalized local Morrey spaces and generalized local Campanato spaces, within a context of a general quasimetric measure space. The locality is manifested here by a restriction to a subfamily of involved balls. The structural properties of these spaces and the maximal operators associated to them are studied. In numerous remarks, we relate the developed theory, mostly in the “global” case, to the cases existing in the literature. We also suggest a coherent theory of generalized Morrey and Campanato spaces on open proper subsets ofRn.

GENERALIZED MORREY SPACES OVER NONHOMOGENEOUS METRIC MEASURE SPACES

2016 ◽  
Vol 103 (2) ◽  
pp. 268-278 ◽  
Author(s):  
GUANGHUI LU ◽  
SHUANGPING TAO
Keyword(s):  
Integral Operator ◽  
Measure Space ◽  
Fractional Integral ◽  
Morrey Spaces ◽  
Marcinkiewicz Integral ◽  
Metric Measure Spaces ◽  
Generalized Morrey Spaces ◽  
Measure Spaces ◽  
Definition Of ◽  
Marcinkiewicz Integral Operator

Let $({\mathcal{X}},d,\unicode[STIX]{x1D707})$ be a nonhomogeneous metric measure space satisfying the so-called upper doubling and the geometric doubling conditions. In this paper, the authors give the natural definition of the generalized Morrey spaces on $({\mathcal{X}},d,\unicode[STIX]{x1D707})$, and then investigate some properties of the maximal operator, the fractional integral operator and its commutator, and the Marcinkiewicz integral operator.

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.

scholarly journals Fractional Maximal Functions in Metric Measure Spaces

2013 ◽  
Vol 1 ◽  
pp. 147-162 ◽  
Author(s):  
Toni Heikkinen ◽  
Juha Lehrbäck ◽  
Juho Nuutinen ◽  
Heli Tuominen
Keyword(s):  
Maximal Function ◽  
Lipschitz Function ◽  
Measure Space ◽  
Continuous Functions ◽  
Maximal Operators ◽  
Metric Measure Spaces ◽  
Hölder Continuous ◽  
Fractional Maximal Function ◽  
Measure Spaces ◽  
Hölder Continuous Functions

Abstract We study the mapping properties of fractional maximal operators in Sobolev and Campanato spaces in metric measure spaces. We show that, under certain restrictions on the underlying metric measure space, fractional maximal operators improve the Sobolev regularity of functions and map functions in Campanato spaces to Hölder continuous functions. We also give an example of a space where fractional maximal function of a Lipschitz function fails to be continuous.

scholarly journals On relations between weak and strong type inequalities for modified maximal operators on non-doubling metric measure spaces

2019 ◽  
Vol 31 (3) ◽  
pp. 785-801
Author(s):  
Dariusz Kosz
Keyword(s):  
Special Class ◽  
Measure Space ◽  
Necessary Conditions ◽  
Metric Measure Space ◽  
Analogous Problem ◽  
Maximal Operators ◽  
Metric Measure Spaces ◽  
Strong Type ◽  
Measure Spaces

Abstract In this article, we investigate a special class of non-doubling metric measure spaces in order to describe the possible configurations of {P_{k,{\mathrm{s}}}^{{\mathrm{c}}}} , {P_{k,{\mathrm{s}}}} , {P_{k,{\mathrm{w}}}^{{\mathrm{c}}}} and {P_{k,{\mathrm{w}}}} , the sets of all {p\in[1,\infty]} for which the weak and strong type {(p,p)} inequalities hold for the centered and non-centered modified Hardy–Littlewood maximal operators {M^{{\mathrm{c}}}_{k}} and {M_{k}} , {k\geq 1} . For any fixed k we describe the necessary conditions that {P_{k,{\mathrm{s}}}^{{\mathrm{c}}}} , {P_{k,{\mathrm{s}}}} , {P_{k,{\mathrm{w}}}^{{\mathrm{c}}}} and {P_{k,{\mathrm{w}}}} must satisfy in general and illustrate each admissible configuration with a properly chosen non-doubling metric measure space. We also give some partial results related to an analogous problem stated for varying k.

Generalized Riesz potentials of functions in Morrey spaces L(1,ϕ;κ)(G) over non-doubling measure spaces

2020 ◽  
Vol 32 (2) ◽  
pp. 339-359 ◽  
Author(s):  
Yoshihiro Sawano ◽  
Masaki Shigematsu ◽  
Tetsu Shimomura
Keyword(s):  
Measure Space ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Doubling Measure ◽  
Doubling Condition ◽  
Riesz Potentials ◽  
General Measure ◽  
Open Set ◽  
Measure Spaces

AbstractThis paper proves the boundedness of the generalized Riesz potentials {I_{\rho,\mu,\tau}f} of functions in the Morrey space {L^{(1,\varphi;\kappa)}(G)} over a general measure space X, with G a bounded open set in X (or G is {X)}, as an extension of earlier results. The modification parameter τ is introduced for the purpose of including the case where the underlying measure does not satisfy the doubling condition. What is new in the present paper is that ρ depends on {x\in X}. An example in the end of this article convincingly explains why the modification parameter τ must be introduced.

Commutators of sublinear operators in grand morrey spaces

2019 ◽  
Vol 56 (2) ◽  
pp. 211-232
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi ◽  
Humberto Rafeiro
Keyword(s):  
Integral Operators ◽  
Morrey Spaces ◽  
Doubling Measure ◽  
Maximal Operators ◽  
Metric Measure Spaces ◽  
Potential Operators ◽  
Sublinear Operators ◽  
Measure Spaces

Abstract In this paper we establish the boundedness of commutators of sublinear operators in weighted grand Morrey spaces. The sublinear operators under consideration contain integral operators such as Hardy-Littlewood and fractional maximal operators, Calderón-Zygmund operators, potential operators etc. The operators and spaces are defined on quasi-metric measure spaces with doubling measure.

scholarly journals Geometry and dynamics of admissible metrics in measure spaces

2013 ◽  
Vol 11 (3) ◽  
Author(s):  
Anatoly Vershik ◽  
Pavel Zatitskiy ◽  
Fedor Petrov
Keyword(s):  
Invariant Measure ◽  
Wide Class ◽  
Lebesgue Space ◽  
Measure Space ◽  
Asymptotic Properties ◽  
Ergodic Transformation ◽  
Measure Spaces ◽  
Admissible Metric ◽  
Discreteness Criterion ◽  
Uniformly Bounded

AbstractWe study a wide class of metrics in a Lebesgue space, namely the class of so-called admissible metrics. We consider the cone of admissible metrics, introduce a special norm in it, prove compactness criteria, define the ɛ-entropy of a measure space with an admissible metric, etc. These notions and related results are applied to the theory of transformations with invariant measure; namely, we study the asymptotic properties of orbits in the cone of admissible metrics with respect to a given transformation or a group of transformations. The main result of this paper is a new discreteness criterion for the spectrum of an ergodic transformation: we prove that the spectrum is discrete if and only if the ɛ-entropy of the averages of some (and hence any) admissible metric over its trajectory is uniformly bounded.

Boundedness of some sublinear operators and their commutators on generalized local Morrey spaces

2017 ◽  
Vol 63 (11) ◽  
pp. 1620-1641
Author(s):  
A. S. Balakishiyev ◽  
E. A. Gadjieva ◽  
F. Gürbüz ◽  
A. Serbetci
Keyword(s):  
Morrey Spaces ◽  
Sublinear Operators ◽  
Local Morrey Spaces
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