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 Estimates for Multilinear Commutators of Generalized Fractional Integral Operators on Weighted Morrey Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-11
Author(s):  
Sha He ◽  
Taotao Zheng ◽  
Xiangxing Tao
Keyword(s):  
Morrey Spaces ◽  
Fractional Integrals ◽  
Gaussian Kernel ◽  
Bmo Space ◽  
Laplacian Operator ◽  
Fractional Integral Operators ◽  
Weighted Bmo ◽  
Weighted Morrey Spaces ◽  
Integrable Functions ◽  
Multilinear Commutators

LetLbe the infinitesimal generator of an analytic semigroup onL2(Rn)with Gaussian kernel bounds, and letL-α/2be the fractional integrals ofLfor0<α<n. Assume thatb→=(b1,b2,…,bm)is a finite family of locally integrable functions; then the multilinear commutators generated byb→andL-α/2are defined byLb→-α/2f=[bm,…,[b2,[b1,L-α/2]],…]f. Assume thatbjbelongs to weighted BMO space,j=1,2,…,m; the authors obtain the boundedness ofLb→-α/2on weighted Morrey spaces. As a special case, whenL=-Δis the Laplacian operator, the authors also obtain the boundedness of the multilinear fractional commutatorIαb→on weighted Morrey spaces. The main results in this paper are substantial improvements and extensions of some known results.

scholarly journals Estimates for Fractional Integral Operators and Linear Commutators on Certain Weighted Amalgam Spaces

2020 ◽  
Vol 2020 ◽  
pp. 1-25 ◽  
Author(s):  
Hua Wang
Keyword(s):  
Fractional Integral ◽  
Weak Type ◽  
Integral Operators ◽  
Fractional Integrals ◽  
Gaussian Kernel ◽  
Strong Type ◽  
Fractional Integral Operators ◽  
Weak Type Estimates ◽  
Kernel Bounds ◽  
Amalgam Spaces

In this paper, we first introduce some new classes of weighted amalgam spaces. Then, we give the weighted strong-type and weak-type estimates for fractional integral operators Iγ on these new function spaces. Furthermore, the weighted strong-type estimate and endpoint estimate of linear commutators b,Iγ generated by b and Iγ are established as well. In addition, we are going to study related problems about two-weight, weak-type inequalities for Iγ and b,Iγ on the weighted amalgam spaces and give some results. Based on these results and pointwise domination, we can prove norm inequalities involving fractional maximal operator Mγ and generalized fractional integrals ℒ−γ/2 in the context of weighted amalgam spaces, where 0<γ<n and L is the infinitesimal generator of an analytic semigroup on L2Rn with Gaussian kernel bounds.

scholarly journals Multilinear Singular and Fractional Integral Operators on Weighted Morrey Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Hua Wang ◽  
Wentan Yi
Keyword(s):  
Fractional Integral ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Fractional Integrals ◽  
Fractional Integral Operators ◽  
Weighted Morrey Spaces ◽  
Multiple Weights ◽  
Multilinear Fractional Integrals

We will study the boundedness properties of multilinear Calderón-Zygmund operators and multilinear fractional integrals on products of weighted Morrey spaces with multiple weights.

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 Operators with Gaussian kernel bounds on mixed Morrey spaces

Filomat ◽  
2019 ◽  
Vol 33 (16) ◽  
pp. 5219-5230 ◽  
Author(s):  
Francesca Anceschi ◽  
Christopher Goodrich ◽  
Andrea Scapellato
Keyword(s):  
Analytic Semigroup ◽  
Morrey Spaces ◽  
Gaussian Kernel ◽  
Fractional Operator ◽  
Lipschitz Spaces ◽  
Kernel Bounds

Let L be an analytic semigroup on L2(Rn) with Gaussian kernel bound, and let L-?/2 be the fractional operator associated to L for 0 < ? < n. In this paper, we prove some boundedness properties for the commutator [b,L-?/2] on Mixed Morrey spaces Lq,? (0,T,Lp,?(Rn)), when b belongs to BMO(Rn) or to suitable homogeneous Lipschitz spaces.

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.

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