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 Calder′on Zygmund operators on variableexponent Morrey spaces over domains

The boundedness of multilinear singular integrals of Calder′on-Zygmund type onproduct of variable exponent Lebesgue spaces over both bounded and unbounded domains areobtained. Further more, the boundedness for this type multilinear operators on product ofvariable exponent Morrey spaces over domains is shown in the paper.

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.

Some Estimates for Commutators of Fractional Integrals Associated to Operators with Gaussian Kernel Bounds on Weighted Morrey Spaces

Let L be the infinitesimal generator of an analytic semigroup on L^2 （R^n） with Gaussian kernel bound, and let L^-α/2 be the fractional integrals of L for 0 〈 α 〈 n. In this paper, we will obtain some boundedness properties of commutators [b, L^-α/2] on weighted Morrey spaces L^p,k（w） when the symbol b belongs to BMO（Rn） or the homogeneous Lipschitz space.

The Fractional Maximal Operator and Marcinkiewicz Integrals Associated with Schr?dinger Operators on Morrey Spaces with Variable Exponent

In this paper, we prove the boundedness of the fractional maximal operator, Hardy-Littlewood maximal operator and marcinkiewicz integrals associated with Schrodinger operator on Morrey spaces with variable exponent.

Boundedness of Operators in Morrey Spaces over Vilenkin Groups

Let G be a locally compact Vilenkin gro up . We will establish the boundedness in Morrey spaces L p,λ (G) for a la rge class of sublinear operators and linear commutators.

Boundedness for Commutators of Calder o′n-Zygmund Operator on Morrey Spaces with Variable Exponent

Our aim in the present paper is to prove the boundedness of commutators on Morrey spaces with variable exponent. In order to obtain the result, we clarify a relation between variable exponent and BMO norms.

Weak Type Estimates for Intrinsic Square Functions on Weighted Morrey Spaces

In this paper, we will obtain the weak type estimates of intrinsic square func- tions including the Lusin area integral, Littlewood-Paley g-function and g^-function on the weighted Morrey spaces L^1,k （w） for 0〈k〈 1 and w ∈ A1.

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 for Maximal Bochner-Riesz Operator and Commutator on Generalized Morrey Space

In this paper,we have obtained the boundedness of maximal Bochner-Riesz operator on generalized Morrey space.Also,it is right for its commutator.

BOUNDS FOR MULTILINEAR OPERATORS UNDER AN INTEGRAL TYPE CONDITION ON MORREY SPACES

In this paper,we study a boundedness property of the Adams type for multilinear fractional integral operators with the multilinear L^(r′,α)-Hörmander condition and their commutators with vector valued BMO functions on a Morrey space and a predual Morrey space.Moreover,we give an endpoint estimate for multilinear fractional integral operators.As an application,we obtain the boundedness of multilinear Fourier multipliers with limited Sobolev regularity on a Morrey space.

DOOB'S INEQUALITY, BURKHOLDER-GUNDY INEQUALITY AND MARTINGALE TRANSFORMS ON MARTINGALE MORREY SPACES

We introduce the martingale Morrey spaces built on Banach function spaces. We establish the Doob＇s inequality, the Burkholder-Gundy inequality and the boundedness of martingale transforms for our martingale Morrey spaces. We also introduce the martingale block spaces. By the Doob＇s inequality on martingale block spaces, we obtain the Davis＇ decompositions for martingale Morrey spaces.

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.

Boundedness for the Singular Integral with Variable Kernel and Fractional Differentiation on Weighted Morrey Spaces

Let T be the singular integral operator with variable kernel, T＊ be the adjoint of T and T# be the pseudo-adjoint of T. Let TIT2 be the product of T1 and T2, T1 o T2 be the pseudo product of T1 and T2. In this paper, we establish the boundedness for commutators of these operators and the fractional differentiation operator D^γ on the weighted Morrey spaces.

Toeplitz Type Operator Associated to Singular Integral Operator with Variable Kernel on Weighted Morrey Spaces

Suppose T^k,l and T^k,2 are singular integrals with variable kernels and mixed homogeneity or ±I （the identity operator）. Denote the Toeplitz type operator by T^b=k=1∑^QT^k,1M^bT^k,2 where M^bf= bf. In this paper, the boundedness of Tb on weighted Morrey space are obtained when b belongs to the weighted Lipschitz function space and weighted BMO function space, respectively.

A CHARACTERIZATION FOR FRACTIONAL INTEGRALS ON GENERALIZED MORREY SPACES

This paper concerns with the fractional integrals, which are also known as the Riesz potentials. A characterization for the boundedness of the fractional integral operators on generalized Morrey spaces will be presented. Our results can be viewed as a refinement of Nakai＇s.

The Boundedness of the Singular Integral Operator with Variable Calder′on-Zygmund Kernel on Weighted Morrey Spaces

In this paper, we will study the boundedness of the singular integral operator with variable Calder′on-Zygmund kernel on the weighted Morrey spaces Lp,κ(ω) for q′≤ p < ∞and 0 < κ < 1. Furthermore, the boundedness for the commutator with BMO functions is also obtained.

ON BOUNDEDNESS PROPERTY OF SINGULAR INTEGRAL OPERATORS ASSOCIATED TO A SCHRODINGER OPERATOR IN A GENERALIZED MORREY SPACE AND APPLICATIONS

In this paper,we provide the boundedness property of the Riesz transforms associated to the Schrodinger operator■=-Δ+V in a new weighted Morrey space which is the generalized version of many previous Morrey type spaces.The additional potential V considered in this paper is a non-negative function satisfying the suitable reverse Holder's inequality.Our results are new and general in many cases of problems.As an application of the boundedness property of these singular integral operators,we obtain some regularity results of solutions to Schrodinger equations in the new Morrey space.

Toeplitz Operator Related to Singular Integral with Non-Smooth Kernel on Weighted Morrey Space

Let T1 be a singular integral with non-smooth kernel or ±I, let T2 and T4 be the linear operators and let T3 = ± I. Denote the Toeplitz type operator by Tb = T1M^b Ia T2 ＋ T3IaMbT4,where M^bf = bf, and Ib is the fractional integral operator. In this paper, we investigate the boundedness of the operator Tb on the weighted Morrey space when b belongs to the weighted BMO space.