Admissible Sublinear Singular Operators and Generalized Orlicz-Morrey Spaces (original) (raw)
Φ-Admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces
Journal of Inequalities and Applications, 2014
We study the boundedness of-admissible singular operators and their commutators on vanishing generalized Orlicz-Morrey spaces VM ,ϕ (R n) including their weak versions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator, the Calderón-Zygmund singular integral operator and so on. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on weights ϕ(x, r) without assuming any monotonicity property of ϕ(x, r) on r.
Boundedness of the Maximal and Singular Operators on Generalized Orlicz–Morrey Spaces
Operator Theory: Advances and Applications, 2014
We consider generalized Orlicz-Morrey spaces MΦ,ϕ(R n) including their weak versions. In these generalized spaces we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. In all the cases the conditions for the boundedness are given either in terms of Zygmundtype integral inequalities on ϕ(r) without assuming any monotonicity property of ϕ(r), or in terms of supremal operators, related to ϕ(r).
Boundedness of the Maximal, Potential and Singular Operators in the Generalized Morrey Spaces
Journal of Inequalities and Applications, 2009
We consider generalized Morrey spaces M p,ω R n with a general function ω x, r defining the Morrey-type norm. We find the conditions on the pair ω 1 , ω 2 which ensures the boundedness of the maximal operator and Calderón-Zygmund singular integral operators from one generalized Morrey space M p,ω1 R n to another M p,ω2 R n , 1 < p < ∞, and from the space M 1,ω1 R n to the weak space WM 1,ω2 R n. We also prove a Sobolev-Adams type M p,ω1 R n → M q,ω2 R n-theorem for the potential operators I α. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω 1 , ω 2 , which do not assume any assumption on monotonicity of ω 1 , ω 2 in r. As applications, we establish the boundedness of some Schrödinger type operators on generalized Morrey spaces related to certain nonnegative potentials belonging to the reverse Hölder class. As an another application, we prove the boundedness of various operators on generalized Morrey spaces which are estimated by Riesz potentials.
Maximal, potential and singular operators in vanishing generalized Morrey spaces
Journal of Global Optimization, 2012
We introduce vanishing generalized Morrey spaces V L p,ϕ (), ⊆ R n with a general function ϕ(x, r) defining the Morrey-type norm. Here ⊆ is an arbitrary subset in including the extremal cases = {x 0 }, x 0 ∈ and = , which allows to unify vanishing local and global Morrey spaces. In the spaces V L p,ϕ (R n) we prove the boundedness of a class of sublinear singular operators, which includes Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel. We also prove a Sobolev-Spanne type V L p,ϕ (R n) → V L q,ϕ q p (R n)-theorem for the potential operator I α. The conditions for the boundedness are given in terms of Zygmund-type integral inequalities on ϕ(x, r). No monotonicity type condition is imposed on ϕ(x, r). In case ϕ has quasimonotone properties, as a consequence of the main results, the conditions of the boundedness are also given in terms of the Matuszeska-Orlicz indices of the function ϕ. The proofs are based on pointwise estimates of the modulars defining the vanishing spaces
Collectanea Mathematica, 2015
We study the boundedness of (,)-admissible potential operators and their commutators on vanishing generalized Orlicz-Morrey spaces V M ,ϕ (R n) including their weak versions. These conditions are satisfied by most of the operators in harmonic analysis, such as the Riesz potential, fractional maximal operator and so on. In all the cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities involving the Young functions , and the function ϕ(x, r) defining the space, without assuming any monotonicity property of ϕ(x, r) on r .
Annals of the Alexandru Ioan Cuza University - Mathematics, 2014
In this paper we study the boundedness for a large class of sublinear operators T generated by Calderón-Zygmund operators on generalized weighted Morrey spaces Mp,φ(w) with the weight function w(x) belonging to Muckenhoupt's class Ap. We find the sufficient conditions on the pair (φ1, φ2) which ensures the boundedness of the operator T from one generalized weighted Morrey space Mp,φ 1 (w) to another Mp,φ 2 (w) for p > 1 and from M1,φ 1 (w) to the weak space W M1,φ 2 (w). In all cases the conditions for the boundedness are given in terms of Zygmund-type integral inequalities on (φ1, φ2), which do not assume any assumption on monotonicity of φ1, φ2 in r. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.
Boundedness of a Class of Sublinear Operators and Their Commutators on Generalized Morrey Spaces
Abstract and Applied Analysis, 2011
The authors study the boundedness for a large class of sublinear operatorTgenerated by Calderón-Zygmund operator on generalized Morrey spacesMp,φ. As an application of this result, the boundedness of the commutator of sublinear operatorsTaon generalized Morrey spaces is obtained. In the casea∈BMO(ℝn),1<p<∞andTais a sublinear operator, we find the sufficient conditions on the pair (φ1,φ2) which ensures the boundedness of the operatorTafrom one generalized Morrey spaceMp,φ1to anotherMp,φ2. In all cases, the conditions for the boundedness ofTaare given in terms of Zygmund-type integral inequalities on (φ1,φ2), which do not assume any assumption on monotonicity ofφ1,φ2inr. Conditions of these theorems are satisfied by many important operators in analysis, in particular pseudodifferential operators, Littlewood-Paley operator, Marcinkiewicz operator, and Bochner-Riesz operator.
Annales Academiae Scientiarum Fennicae Mathematica, 2015
We prove the boundedness of the Hardy-Littlewood maximal operator and their commutators with BMO-coefficients in vanishing generalized Orlicz-Morrey spaces V M Φ,ϕ (R n) including weak versions of these spaces. The main advance in comparison with the existing results is that we manage to obtain conditions for the boundedness not in integral terms but in less restrictive terms of supremal operators involving the Young function Φ(u) and the function ϕ(x, r) defining the space. No kind of monotonicity condition on ϕ(x, r) in r is imposed.
Maximal, potential and singular operators
2011
We consider local "complementary" generalized Morrey spaces ∁ M p(•),ω {x 0 } (Ω) in which the p-means of function are controlled over Ω\B(x 0 , r) instead of B(x 0 , r) , where Ω ⊂ R n is a bounded open set, p(x) is a variable exponent, and no monotonicity type conditio is imposed onto the function ω(r) defining the "complementary" Morrey-type norm. In the case where ω is a power function, we reveal the relation of these spaces to weighted Lebesgue spaces. In the general case we prove the boundedness of the Hardy-Littlewood maximal operator and Calderon-Zygmund singular operators with standard kernel, in such spaces. We also prove a Sobolev type ∁ M p(•),ω {x 0 } (Ω) → ∁ M q(•),ω {x 0 } (Ω)-theorem for the potential operators I α(•) , also of variable order. In all the cases the conditions for the boundedness are given it terms of Zygmund-type integral inequalities on ω(r) , which do not assume any assumption on monotonicity of ω(r) .