scholarly journals Variable exponent fractional integrals in the limiting case α(x)p(n) ≡ n on quasimetric measure spaces

2020 ◽  
Vol 27 (1) ◽  
pp. 157-164
Author(s):  
Stefan Samko
Keyword(s):  
Growth Condition ◽  
Lebesgue Space ◽  
Measure Space ◽  
Variable Exponent ◽  
Fractional Integrals ◽  
Variable Order ◽  
Fractional Operator ◽  
Open Set ◽  
Measure Spaces ◽  
Limiting Case

AbstractWe show that the fractional operator {I^{\alpha(\,\cdot\,)}}, of variable order on a bounded open set in Ω, in a quasimetric measure space {(X,d,\mu)} in the case {\alpha(x)p(x)\equiv n} (where n comes from the growth condition on the measure μ), is bounded from the variable exponent Lebesgue space {L^{p(\,\cdot\,)}(\Omega)} into {\mathrm{BMO}(\Omega)} under certain assumptions on {p(x)} and {\alpha(x)}.

A note on vanishing Morrey → VMO result for fractional integrals of variable order

2020 ◽  
Vol 23 (1) ◽  
pp. 298-302 ◽  
Author(s):  
Humberto Rafeiro ◽  
Stefan Samko
Keyword(s):  
Morrey Space ◽  
Morrey Spaces ◽  
Fractional Integrals ◽  
Variable Order ◽  
Fractional Operator ◽  
Limiting Case

AbstractIn the limiting case of Sobolev-Adams theorem for Morrey spaces of variable order we prove that the fractional operator of variable order maps the corresponding vanishing Morrey space into VMO.

scholarly journals A note on Riesz fractional integrals in the limiting case α(x)p(x) ≡ n

2013 ◽  
Vol 16 (2) ◽  
Author(s):  
Stefan Samko
Keyword(s):  
Fractional Integration ◽  
Fractional Integrals ◽  
Variable Order ◽  
Integration Operator ◽  
Hölder Continuous ◽  
Open Set ◽  
Fractional Integration Operator ◽  
Limiting Case

AbstractWe show that the Riesz fractional integration operator I α(·) of variable order on a bounded open set in Ω ⊂ ℝn in the limiting Sobolev case is bounded from L p(·)(Ω) into BMO(Ω), if p(x) satisfies the standard logcondition and α(x) is Hölder continuous of an arbitrarily small order.

Maximal and Potential Operators in Variable Exponent Morrey Spaces

2008 ◽  
Vol 15 (2) ◽  
pp. 195-208 ◽  
Author(s):  
Alexandre Almeida ◽  
Javanshir Hasanov ◽  
Stefan Samko
Keyword(s):  
Variable Exponent ◽  
Morrey Spaces ◽  
Bmo Space ◽  
Sobolev Type ◽  
Potential Operator ◽  
Variable Order ◽  
Potential Operators ◽  
Open Set ◽  
Limiting Case ◽  
Littlewood Maximal Operator

Abstract We prove the boundedness of the Hardy–Littlewood maximal operator on variable Morrey spaces 𝐿𝑝(·), λ(·)(Ω) over a bounded open set Ω ⊂ ℝ𝑛 and a Sobolev type 𝐿𝑝(·), λ(·) → 𝐿𝑞(·), λ(·)-theorem for potential operators 𝐼 α(·), also of variable order. In the case of constant α, the limiting case is also studied when the potential operator 𝐼 α acts into BMO space.

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.

A variable exponent Sobolev theorem for fractional integrals on quasimetric measure spaces

Analysis ◽  
2016 ◽  
Vol 36 (1) ◽  
Author(s):  
Stefan Samko
Keyword(s):  
Variable Exponent ◽  
Fractional Integrals ◽  
Measure Spaces

AbstractWe prove that the coefficients

scholarly journals Estimates for Parameter Littlewood-Paleygκ⁎Functions on Nonhomogeneous Metric Measure Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-12 ◽  
Author(s):  
Guanghui Lu ◽  
Shuangping Tao
Keyword(s):  
Hardy Space ◽  
Lebesgue Space ◽  
Measure Space ◽  
Metric Measure Space ◽  
Doubling Measure ◽  
Metric Measure Spaces ◽  
Type Condition ◽  
Lebesgue Spaces ◽  
Measure Spaces ◽  
Atomic Hardy Space

Let(X,d,μ)be a metric measure space which satisfies the geometrically doubling measure and the upper doubling measure conditions. In this paper, the authors prove that, under the assumption that the kernel ofMκ⁎satisfies a certain Hörmander-type condition,Mκ⁎,ρis bounded from Lebesgue spacesLp(μ)to Lebesgue spacesLp(μ)forp≥2and is bounded fromL1(μ)intoL1,∞(μ). As a corollary,Mκ⁎,ρis bounded onLp(μ)for1<p<2. In addition, the authors also obtain thatMκ⁎,ρis bounded from the atomic Hardy spaceH1(μ)into the Lebesgue spaceL1(μ).

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.

Maximal Operator in Variable Exponent Lebesgue Spaces on Unbounded Quasimetric Measure Spaces

2015 ◽  
Vol 116 (1) ◽  
pp. 5 ◽  
Author(s):  
Tomasz Adamowicz ◽  
Petteri Harjulehto ◽  
Peter Hästö
Keyword(s):  
Maximal Operator ◽  
Measure Space ◽  
Variable Exponent ◽  
Hölder Condition ◽  
Lebesgue Spaces ◽  
General Measure ◽  
Radon Measures ◽  
Measure Spaces ◽  
Littlewood Maximal Operator ◽  
Special Case

We study the Hardy-Littlewood maximal operator $M$ on $L^{p({\cdot})}(X)$ when $X$ is an unbounded (quasi)metric measure space, and $p$ may be unbounded. We consider both the doubling and general measure case, and use two versions of the $\log$-Hölder condition. As a special case we obtain the criterion for a boundedness of $M$ on $L^{p({\cdot})}({\mathsf{R}^n},\mu)$ for arbitrary, possibly non-doubling, Radon measures.

Trace inequalities for fractional integrals in mixed norm grand lebesgue spaces

2020 ◽  
Vol 23 (5) ◽  
pp. 1452-1471
Author(s):  
Vakhtang Kokilashvili ◽  
Alexander Meskhi
Keyword(s):  
Riesz Potential ◽  
Fractional Integrals ◽  
Mixed Norm ◽  
Lebesgue Spaces ◽  
Fractional Integral Operators ◽  
Grand Lebesgue Spaces ◽  
Measure Spaces ◽  
Necessary And Sufficient ◽  
Bounded Sets ◽  
Trace Inequalities

Abstract D. Adams type trace inequalities for multiple fractional integral operators in grand Lebesgue spaces with mixed norms are established. Operators under consideration contain multiple fractional integrals defined on the product of quasi-metric measure spaces, and one-sided multiple potentials. In the case when we deal with operators defined on bounded sets, the established conditions are simultaneously necessary and sufficient for appropriate trace inequalities. The derived results are new even for multiple Riesz potential operators defined on the product of Euclidean spaces.

Sign in / Sign up

Export Citation Format

Share Document