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.

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.

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.

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.

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.

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.

Weighted Lipschitz Estimate for Commutator of Bochner-Riesz Operators on Weighted Morrey Spaces

In this paper, we will use a method of sharp maximal function approach to show the boundedness of commutator [b,TR^δ] by Bochner-Riesz operators and the function b on weighted Morrey spaces L^p,k （ω） under appropriate conditions on the weight w, where b belongs to Lipschitz space or weighted Lipschitz space.

Boundedness for Commutators of Approximate Identities on Weighted Morrey Spaces

The aim of this paper is to set up the weighted norm inequalities for commutators generated by approximate identities from weighted Lebesgue spaces into weighted Morrey spaces

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.

Variation operators for singular integrals and their commutators on weighted Morrey spaces and Sobolev spaces

In this paper,we investigate the boundedness and compactness for variation operators of CalderónZygmund singular integrals and their commutators on weighted Morrey spaces and Sobolev spaces.To be precise,letρ>2 and K be a standard Calderón-Zygmund kernel.Denote by V_(ρ)(T_(K))and V_(ρ)(T_(K,b)^(m))(m≥1)theρ-variation operators of Calderón-Zygmund singular integrals and their m-th iterated commutators,respectively.By assuming that V_(ρ)(T_(K))satisfies an a priori estimate,i.e.,the map V_(ρ)(T_(K)):L^(p0)(R^(n))→L^(p0)(R^(n))is bounded for some p0∈(1,∞),the bounds for V_(ρ)(T_(K))and V_(ρ)(T_(K,b)^(m))on weighted Morrey spaces and Sobolev spaces are established.Meanwhile,the compactness properties of V_(ρ)(T_(K,b)^(m))on weighted Lebesgue and Morrey spaces are also discussed.As applications,the corresponding results for the Hilbert transform,the Hermite Riesz transform,Riesz transforms and rough singular integrals as well as their commutators on the above function spaces are presented.

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.

Bilinear Pseudo-Differential Operator and Its Commutator on Generalized Fractional Weighted Morrey Spaces

The aim of this paper is to establish the boundedness of bilinear pseudodifferential operator T_(σ) and its commutator[b_(1),b_(2),T_(σ)]generated by T_(σ) and b_(1),b_(2) BMO(R^(n))on generalized fractional weighted Morrey spaces L^(p,η,φ)(w).Under assumption that a weight satisfies a certain condition,the authors prove that Ts is bounded from products of spaces L^(p1,η1,φ)(w1)L^(p2,η2,φ)(w2)into spaces L^(p,η,φ)(w),where w=(w_(1),w_(2)) A_(P),P=(p1,p2),η=η1+η2 and 1/p=1/p_(1)+1/p_(2) with p_(1),p_(2)(1,∞).Furthermore,the authors show that the[b1,b2,T_(σ)]is bounded from products of generalized fractional Morrey spaces L^(p1,η1,φ)(R^(n))L^(p2,η2,φ)(R^(n))into L^(p,η,φ)(R^(n)).As corollaries,the boundedness of the T_(σ) and[b_(1),b_(2),T_(σ)]on generalized weighted Morrey spaces L^(p,φ)(w)and on generalized Morrey spaces L^(p,φ)(R^(n))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.

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.

The Weighted Morrey Boundedness of Multilinear Singular Integral Operators on RD-Spaces

An RD-space X is a space of homogeneous type in the sense of Coifman and Weiss,which is equipped with a measure satisfying an additional reverse doubling property.In this paper we study the boundedness of multilinear singular integral operators in weighted Morrey spaces within the framework of RD-spaces.

ANOTHER CHARACTERIZATIONS OF MUCKENHOUPT A_p CLASS

This manuscript addresses Muckenhoupt Ap weight theory in connection to Mor- rey and BMO spaces. It is proved that a; belongs to Muckenhoupt Ap class, if and only if Hardy-Littlewood maximal function M is bounded from weighted Lebesgue spaces LP（w） to weighted Morrey spaces Mpq（ω） for 1 〈 q 〈 p 〈 ∞. As a corollary, if M is （weak） bounded on Mpq（ω）, then ω∈Ap. The Ap condition also characterizes the boundedness of the Riesz transform Rj and convolution operators Tε on weighted Morrey spaces. Finally, we show that ω∈Ap if and only if ω∈BMOp＇ （ω） for 1 ≤ p 〈 ∞ and 1/p ＋ 1/p＇ = 1.

THE EXISTENCE OF A NONTRIVIAL WEAK SOLUTION TO A DOUBLE CRITICAL PROBLEM INVOLVING A FRACTIONAL LAPLACIAN IN R^N WITH A HARDY TERM

In this paper,we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:(−Δ)s u−γu|x|2s=|u|2∗s(β)−2 u|x|β+[Iμ∗Fα(⋅,u)](x)fα(x,u),u∈H˙s(R n),(0.1)(1)where s∈(0,1),0≤α,β<2s<n,μ∈(0,n),γ<γH,Iμ(x)=|x|−μ,Fα(x,u)=|u(x)|2#μ(α)|x|δμ(α),fα(x,u)=|u(x)|2#μ(α)−2 u(x)|x|δμ(α),2#μ(α)=(1−μ2n)⋅2∗s(α),δμ(α)=(1−μ2n)α,2∗s(α)=2(n−α)n−2s andγH=4 sΓ2(n+2s4)Γ2(n−2s4).We show that problem(0.1)admits at least a weak solution under some conditions.To prove the main result,we develop some useful tools based on a weighted Morrey space.To be precise,we discover the embeddings H˙s(R n)↪L 2∗s(α)(R n,|y|−α)↪L p,n−2s2 p+pr(R n,|y|−pr),(0.2)(2)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α))and r=α2∗s(α).We also establish an improved Sobolev inequality,(∫R n|u(y)|2∗s(α)|y|αdy)12∗s(α)≤C||u||θH˙s(R n)||u||1−θL p,n−2s2 p+pr(R n,|y|−pr),∀u∈H˙s(R n),(0.3)(3)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α)),r=α2∗s(α),0<max{22∗s(α),2∗s−12∗s(α)}<θ<1,2∗s=2nn−2s and C=C(n,s,α)>0 is a constant.Inequality(0.3)is a more general form of Theorem 1 in Palatucci,Pisante[1].By using the mountain pass lemma along with(0.2)and(0.3),we obtain a nontrivial weak solution to problem(0.1)in a direct way.It is worth pointing out that(0.2)and 0.3)could be applied to simplify the proof of the existence results in[2]and[3].

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.

Estimates for Vector-valued Commutators on Weighted Morrey Space and Applications

In this paper, we introduce weighted vector-valued Morrey spaces and obtain some estimates for vector-valued commutators on these spaces. Applications to CalderSn-Zygmund singular integral operators, oscillatory singular integral operators and parabolic difference equations are considered.

Boundedness for a Class of Operators on Weighted Morrey Space with RD-measure

this paper,we study a class of sublinear operators and their commutators with a weighted BMO function.We first give the definition of a weighted Morrey space Lpo(X)where X is an RD-measure and w is the weight function.The weighted Morrey spaces arise from studying the local behavior of solutions to certain partial differential equations.We will show that the aforementioned class of operators and their communtators with a weighted BMO function are bounded in the weighted Morrey space LPpto(X)provided that the weight function w belongs to the Ap(μ)-class and satisfiesthereverseHolder'scondition.