BOUNDEDNESS OF FRACTIONAL MAXIMAL OPERATOR AND THEIR HIGHER ORDER COMMUTATORS IN GENERALIZED MORREY SPACES ON CARNOT GROUPS

In the article we consider the fractional maximal operator Mα, 0 ≤α 〈 Q on any Carnot group G （i.e., nilpotent stratified Lie group） in the generalized Morrey spaces Mp,φ（G）, where Q is the homogeneous dimension of G. We find the conditions on the pair （φ1, φ2） which ensures the boundedness of the operator Ms from one generalized Morrey space Mp,φ1 （G） to another Mq,φ2 （G）, 1. 〈 p ≤q 〈 ∞. 1/p - 1/q = α/Q, and from the space M1,φ1 （G） to the weak space Wq,φ2 （G）, 1 〈 q 〈 ∞, 1 - 1/q = α/Q. Also find conditions on the φ which ensure the Adams type boundedness of the Ms from M α （G） from Mp,φ^1/p（G）to Mq,φ^1/q（G） for 1 〈p〈q〈∞ and fromM1,φ（G） toWMq,φ^1/q（G）for 1〈q〈∞. In the case b ∈ BMO（G） and 1 〈 p 〈 q 〈 ∞, find the sufficient conditions on the pair （φ1, φ2） which ensures the boundedness of the kth-order commutator operator Mb,α,k from Mp,φ1 （G） to Mq,φ2（G） with 1/p - 1/q = α/Q. Also find the sufficient conditions on the φ which ensures the boundedness of the operator Mb,α,k from Mp,φ^1/p（G） tom Mp,φ^1/p （G） for 1 〈p〈q〈∞. In all the cases the conditions for the boundedness of Mα are given it terms of supremaltype inequalities on （φ1, φ2） and φ , which do not assume any assumption on monotonicity of （φ1, φ2） and φ in r. As applications we consider the SchrSdinger operator -△G ＋ V on G, where the nonnegative potential V belongs to the reverse Holder class B∞（G）. The MB,φ1 - Mq,φ2 estimates for the operators V^γ（-△G ＋ V）^-β and V^γ△↓G（-△G ＋ V）^-β are obtained.

Boundedness of multilinear operators on generalized Morrey spaces

In this paper, the authors prove the boundedness of the multilinear maximal func- tions, multilinear singular integrals and multilinear Riesz potential on the product generalized Rn Rn Morrey spaces Mp1,ωw1 （Rn）×…×Mpm,ω （Rn） respectively. The main theorems of this paper extend some known results.

BOUNDEDNESS FOR THE COMMUTATOR OF FRACTIONAL INTEGRAL ON GENERALIZED MORREY SPACE IN NONHOMOGENEOUS SPACE

In this paper, we will establish the boundedness of the commutator generated by fractional integral operator and RBMO（μ） function on generalized Morrey space in the non-homogeneous space.

Boundedness of Some Multilinear Operators on Generalized Morrey Spaces L^（p,ω）

In this paper,the author study the boundedness of some multilinear operators on generalized Morrey spaces Lp,ω.They are the generalization of the corresponding results about commutators in [6].Even then,from these results,we can concluded the cases of high order commutators.

Singular Integrals and Commutators in Generalized Morrey Spaces

We consider a singular an integral operator K with a variable Calderón-Zygmund type kernel k（x; ξ）, x ∈R^n, ξ∈ R^n／{0}, satisfying a mixed homogeneity condition of the form k（x; μ^α1ξ1,..., μ^αnξn） =μ^-∑i=1^n α≥1 k（x, ξ）, αi≥ 1 and μ 〉 0. The continuity of this operator in L^（^＇＇） is well studied by Fabes and Rivière. Our goal is to extend their result to generalized Morrey spaces L^p,ω（R^n）, p ∈ （1, ∞） with a weight w satisfying suitable dabbling and integral conditions. A special attention is paid to the commutator C[α, k]=Kα- αK with the operator of multiplication by BMO functions.

Parabolic Equations with VMO Coefficients in Generalized Morrey Spaces

In this paper, the authors establish the regularity in generalized Morrey spaces of solutions to parabolic equations with VMO coefficients by means of the theory of singular integrals and linear commutators.

Boundedness of the Anisotropic Maximal and Anisotropic Singular Integral Operators in Generalized Morrey Spaces

In this paper we give the conditions on the pair （~1, W2） which ensures the boundedness of the anisotropic maximal operator and anisotropic singular integral operators from one generalized Morrey space Mp,w1 to another Mp,w2, 1 〈 p 〈 ∞, and from the space M1,w1 to the weak space M1,w2.

Commutators of Parabolic Singular Integrals on the Generalized Morrey Spaces

Let [b, T] be the commutator of parabolic singular integral T. In this paper, the authors prove that the boundedness of [b, T] on the generalized Morrey spaces implies b ∈ BMO（Rn, ρ）. The results in this paper improve and extend the Komori and Mizuhara＇s results.

Weighted Estimates of Variable Kernel Fractional Integral and Its Commutators on Vanishing Generalized Morrey Spaces with Variable Exponent

In this paper,the authors obtain the boundedness of the fractional integral operators with variable kernels on the variable exponent generalized weighted Morrey spaces and the variable exponent vanishing generalized weighted Morrey spaces.And the corresponding commutators generated by BMO function are also considered.

Norm Inequalities for Fractional Integral Operators on Generalized Weighted Morrey Spaces

Considering a class of operators which include fractional integrals related to operators with Gaussian kernel bounds, the fractional integral operators with rough kernels and fractional maximal operators with rough kernels as special cases, we prove that if these operators are bounded on weighted Lebesgue spaces and satisfy some local pointwise control, then these operators and the commutators of these operators with a BMO functions are also bounded on generalized weighted Morrey spaces.

INTERIOR ESTIMATES IN MORREY SPACES FOR SOLUTIONS OF ELLIPTIC EQUATIONS AND WEIGHTED BOUNDEDNESS FOR COMMUTATORS OF SINGULAR INTEGRAL OPERATORS

It is proved that, for the nondivergence elliptic equations Σi,jn=1aijuxixj=f, if f belongs to the generalized Morrey spaces Lp, (w), then uxixj ∈ Lp, (w), where u is the W2,p-solution of the equations. In order to obtain this, the author first establish the weighted boundedness for the commutators of some singular integral operators on Lp, (w).

MULTILINEAR CALDERóN-ZYGMUND OPERATOR ON MORREY TYPE SPACES——In Memory of Professor Yongsheng Sun

In this paper, the authors study the boundedness of a class of multi-sublinear operators on the product of Morrey, Herz-Morrey and generalized Morrey spaces, respectively. As their special cases, the corresponding results of multilinear Galderón-gygmund operator can be obtained.

Gradient Estimates for Parabolic Equations in Generalized Weighted Morrey Spaces

We consider the Cauchy-Dirichlet problem for linear divergence form parabolic operators in bounded Reifenberg flat domain. The coefficients supposed to be only measurable in one of the space variables and small BMO with respect to the others. We obtain Calderon-Zygmund type estimate for the gradient of the solution in generalized weighted Morrey spaces with Muckenhoupt weight.

Generalized Weighted Morrey Estimates for Marcinkiewicz Integrals with Rough Kernel Associated with Schrodinger Operator and Their Commutators

Let L =-△+V(x) be a Schrodinger operator, where △ is the Laplacian on R^n,while nonnegative potential V(x) belonging to the reverse Holder class. The aim of this paper is to give generalized weighted Morrey estimates for the boundedness of Marcinkiewicz integrals with rough kernel associated with Schrodinger operator and their commutators.Moreover, the boundedness of the commutator operators formed by BMO functions and Marcinkiewicz integrals with rough kernel associated with Schrodinger operators is discussed on the generalized weighted Morrey spaces. As its special cases, the corresponding results of Marcinkiewicz integrals with rough kernel associated with Schrodinger operator and their commutators have been deduced, respectively. Also, Marcinkiewicz integral operators, rough Hardy-Littlewood(H-L for short) maximal operators, Bochner-Riesz means and parametric Marcinkiewicz integral operators which satisfy the conditions of our main results can be considered as some examples.