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.

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.

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.

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.

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.