scholarly journals Necessary and Sufficient Conditions for Boundedness of Commutators of the General Fractional Integral Operators on Weighted Morrey Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-14
Author(s):  
Zengyan Si ◽  
Fayou Zhao
Keyword(s):  
Fractional Integral ◽  
Sufficient Conditions ◽  
Critical Index ◽  
Multiplication Operator ◽  
Hölder Condition ◽  
Fractional Integral Operators ◽  
Weighted Morrey Space ◽  
Weighted Morrey Spaces ◽  
Reverse Hölder ◽  
Necessary And Sufficient

We prove thatbis inLipβ(ω)if and only if the commutator[b,L-α/2]of the multiplication operator byband the general fractional integral operatorL-α/2is bounded from the weighted Morrey spaceLp,k(ω)toLq,kq/p(ω1-(1-α/n)q,ω), where0<β<1,0<α+β<n,1<p<n/(α+β),1/q=1/p-(α+β)/n,0≤k<p/q,ωq/p∈A1,andrω>(1-k)/(p/(q-k)), and hererωdenotes the critical index ofωfor the reverse Hölder condition.

scholarly journals Mixed-Norm Amalgam Spaces and Their Predual

Symmetry ◽  
2022 ◽  
Vol 14 (1) ◽  
pp. 74
Author(s):  
Houkun Zhang ◽  
Jiang Zhou
Keyword(s):  
Duality Theory ◽  
Fractional Integral ◽  
Necessary Conditions ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Mixed Norm ◽  
Fractional Integral Operators ◽  
Primal Space ◽  
Necessary And Sufficient ◽  
Amalgam Spaces

In this paper, we introduce mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) and prove the boundedness of maximal function. Then, the dilation argument obtains the necessary and sufficient conditions of fractional integral operators’ boundedness. Furthermore, the strong estimates of linear commutators [b,Iγ] generated by b∈BMO(Rn) and Iγ on mixed-norm amalgam spaces (Lp→,Ls→)α(Rn) are established as well. In order to obtain the necessary conditions of fractional integral commutators’ boundedness, we introduce mixed-norm Wiener amalgam spaces (Lp→,Ls→)(Rn). We obtain the necessary and sufficient conditions of fractional integral commutators’ boundedness by the duality theory. The necessary conditions of fractional integral commutators’ boundedness are a new result even for the classical amalgam spaces. By the equivalent norm and the operators Str(p)(f)(x), we study the duality theory of mixed-norm amalgam spaces, which makes our proof easier. In particular, note that predual of the primal space is not obtained and the predual of the equivalent space does not mean the predual of the primal space.

Compactness criteria for fractional integral operators

2019 ◽  
Vol 22 (5) ◽  
pp. 1269-1283 ◽  
Author(s):  
Vakhtang Kokilashvili ◽  
Mieczysław Mastyło ◽  
Alexander Meskhi
Keyword(s):  
Integral Operator ◽  
Measure Space ◽  
Fractional Integral ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Bounded Domains ◽  
Lebesgue Spaces ◽  
Fractional Integral Operators ◽  
Rectifiable Curves ◽  
Necessary And Sufficient

Abstract We establish necessary and sufficient conditions for the compactness of fractional integral operators from Lp(X, μ) to Lq(X, μ) with 1 < p < q < ∞, where μ is a measure on a quasi-metric measure space X. As an application we obtain criteria for the compactness of fractional integral operators defined in weighted Lebesgue spaces over bounded domains of the Euclidean space ℝn with the Lebesgue measure, and also for the fractional integral operator associated to rectifiable curves of the complex plane.

scholarly journals Commutator Theorems for Fractional Integral Operators on Weighted Morrey Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Zhiheng Wang ◽  
Zengyan Si
Keyword(s):  
Analytic Semigroup ◽  
Sufficient Conditions ◽  
Morrey Spaces ◽  
Fractional Integrals ◽  
Gaussian Kernel ◽  
Fractional Integral Operators ◽  
Weighted Morrey Spaces ◽  
Locally Integrable Function ◽  
Kernel Bounds ◽  
Necessary And Sufficient

LetLbe the infinitesimal generator of an analytic semigroup onL2(Rn)with Gaussian kernel bounds, and letL-α/2be the fractional integrals ofLfor0<α<n. For any locally integrable functionb, the commutators associated withL-α/2are defined by[b,L-α/2](f)(x)=b(x)L-α/2(f)(x)-L-α/2(bf)(x). Whenb∈BMO(ω)(weightedBMOspace) orb∈BMO, the authors obtain the necessary and sufficient conditions for the boundedness of[b,L-α/2]on weighted Morrey spaces, respectively.

scholarly journals Riemann-Liouville and Higher Dimensional Hardy Operators for NonNegative Decreasing Function inLp(·)Spaces

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Muhammad Sarwar ◽  
Ghulam Murtaza ◽  
Irshaad Ahmed
Keyword(s):  
Integral Operator ◽  
Variable Exponent ◽  
Fractional Integral ◽  
Sufficient Conditions ◽  
Fractional Integral Operator ◽  
Lebesgue Spaces ◽  
Higher Dimensional ◽  
Necessary And Sufficient ◽  
Decreasing Functions ◽  
Hardy Operators

One-weight inequalities with general weights for Riemann-Liouville transform andn-dimensional fractional integral operator in variable exponent Lebesgue spaces defined onRnare investigated. In particular, we derive necessary and sufficient conditions governing one-weight inequalities for these operators on the cone of nonnegative decreasing functions inLp(x)spaces.

scholarly journals On the Theory of Multilinear Singular Operators with Rough Kernels on the Weighted Morrey Spaces

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Sha He ◽  
Xiangxing Tao
Keyword(s):  
Singular Integral ◽  
Fractional Integral ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Singular Integral Operators ◽  
Multilinear Operators ◽  
Rough Kernels ◽  
Fractional Integral Operators ◽  
Weighted Morrey Spaces ◽  
Maximal Singular Integral

We study some multilinear operators with rough kernels. For the multilinear fractional integral operatorsTΩ,αAand the multilinear fractional maximal integral operatorsMΩ,αA, we obtain their boundedness on weighted Morrey spaces with two weightsLp,κ(u,v)whenDγA∈Λ˙β  (|γ|=m-1)orDγA∈BMO  (|γ|=m-1). For the multilinear singular integral operatorsTΩAand the multilinear maximal singular integral operatorsMΩA, we show they are bounded on weighted Morrey spaces with two weightsLp,κ(u,v)ifDγA∈Λ˙β  (|γ|=m-1)and bounded on weighted Morrey spaces with one weightLp,κ(w)ifDγA∈BMO  (|γ|=m-1)form=1,2.

scholarly journals Estimates of Some Operators on One-Sided Weighted Morrey Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Shaoguang Shi ◽  
Zunwei Fu
Keyword(s):  
Harmonic Analysis ◽  
Fractional Integral ◽  
Morrey Space ◽  
Morrey Spaces ◽  
Weighted Morrey Space ◽  
Weighted Morrey Spaces

A version of one-sided weighted Morrey space is introduced. The boundedness of some classical one-sided operators in harmonic analysis and PDE on these spaces are discussed, including the Riemann-Liouville fractional integral.

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