Generalized Local Morrey Spaces and Fractional Integral Operators with Rough Kernel

Let M ?,? and I ?,? be the fractional maximal and integral operators with rough kernels, where 0 < ? < n. We study the continuity properties of M ?,? and I ?,? on the generalized local Morrey spaces LMp,?{x0}. We prove that the commutators of these operators with local Campanato functions are bounded. Bibliography: 34 titles. © 2013 Springer Science+Business Media New York.Firat University Scientific Research Projects Management Unit: PYO-FEN 4010.13.003 --The research was partially supported by the grant of Ahi Evran University Scientific Research Projects (PYO-FEN 4010.13.003). -

Fractional maximal operator and its commutators in generalized morrey spaces on Heisenberg group

In this paper we study the boundedness of the fractional maximal operator M ? on Heisenberg group H n in the generalized Morrey spaces M p,? (H n ). We shall give a characterization for the strong and weak type Spanne and Adams type boundedness of M ? on the generalized Morrey spaces, respectively. Also we give a characterization for the Spanne and Adams type boundedness of fractional maximal commutator operator M b,? on the generalized Morrey spaces. © 2018, Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan. All rights reserved

Maximal operator in variable exponent generalized morrey spaces on quasi-metric measure space

We consider generalized Morrey spaces on quasi-metric measure spaces , in general unbounded, with variable exponent p(x) and a general function defining the Morrey-type norm. No linear structure of the underlying space X is assumed. The admission of unbounded X generates problems known in variable exponent analysis. We prove the boundedness results for maximal operator known earlier only for the case of bounded sets X. The conditions for the boundedness are given in terms of the so called supremal inequalities imposed on the function , which are weaker than Zygmund-type integral inequalities often used for characterization of admissible functions . Our conditions do not suppose any assumption on monotonicity of in r

Some Characterizations of BMO Spaces via Commutators in Orlicz Spaces on Stratified Lie Groups

In the paper we study the fractional maximal commutators Mb,α and the commutators of the fractional maximal operator [b, Mα] in the Orlicz spaces LΦ(G) on any stratified Lie group G. We give necessary and sufficient conditions for the boundedness of the operators Mb,α and [b, Mα] on Orlicz spaces LΦ(G) when b belongs to BMO(G) spaces, whereby some new characterizations for certain subclasses of BMO(G) spaces are obtained. © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG

Generalized Sobolev–Morrey estimates for hypoelliptic operators on homogeneous groups

Let G= (RN, ∘ , δλ) be a homogeneous group, Q is the homogeneous dimension of G, X, X1, … , Xm be left invariant real vector fields on G and satisfy Hörmander’s rank condition on RN. Assume that X1, … , Xm(m≤ N- 1) are homogeneous of degree one and X is homogeneous of degree two with respect to the family of dilations (δλ) λ>. Consider the following hypoelliptic operator with drift on GL=∑i,j=1maijXiXj+a0X0,where (aij) is a m× m constant matrix satisfying the elliptic condition in Rm and a≠ 0. In this paper, for this class of operators, we obtain the generalized Sobolev–Morrey estimates by establishing boundedness of a large class of sublinear operators Tα, α∈ [0 , Q) generated by Calderón–Zygmund operators (α= 0) and generated by fractional integral operator (α> 0) on generalized Morrey spaces and proving interpolation results on generalized Sobolev–Morrey spaces on G. The sublinear operators under consideration contain integral operators of harmonic analysis such as Hardy–Littlewood and fractional maximal operators, Calderón–Zygmund operators, fractional integral operators on homogeneous groups, etc. © 2021, The Royal Academy of Sciences, Madrid

Characterizations for the fractional maximal operator and its commutators in generalized weighted Morrey spaces on Carnot groups

In this paper, we shall give a characterization for the strong and weak type Spanne type boundedness of the fractional maximal operator Mα, 0 ≤ α< Q on Carnot group G on generalized weighted Morrey spaces Mp , φ(G, w) , where Q is the homogeneous dimension of G. Also we give a characterization for the Spanne type boundedness of the fractional maximal commutator operator Mb , α on generalized weighted Morrey spaces. © 2020, Springer Nature Switzerland AG

Calderón-Zygmund operators with kernels of Dini’s type on generalized weighted variable exponent Morrey spaces

Let T be a Calderón-Zygmund operator of type ω with ω(t) being nondecreasing and satisfying a kind of Dini’s type condition and let Tb→ be the multilinear commutators of T with BMOm functions. In this paper, we study the boundedness of the operators T and Tb→ on generalized weighted variable exponent Morrey spaces Mp(·),φ(w) with the weight function w belonging to variable Muckenhoupt’s class Ap(·)(Rn). We find the sufficient conditions on the pair (φ1, φ2) with b→ ∈ BMOm(Rn) which ensures the boundedness of the operators T and Tb→ from Mp(·),φ1(w) to Mp(·),φ2(w). © 2021, The Author(s), under exclusive licence to Springer Nature Switzerland AG

Commutators of the fractional maximal function in generalized Morrey spaces on Carnot groups

In this paper, we obtain new results of the Spanne and Adams type boundedness characterization of the fractional maximal commutator operator (Formula presented.), (Formula presented.) on the Carnot group (Formula presented.) (i.e. nilpotent stratified Lie group) in generalized Morrey spaces (Formula presented.) respectively, where Q is the homogeneous dimension of (Formula presented.). © 2020 Informa UK Limited, trading as Taylor & Francis Group

GENERALIZED WEIGHTED SOBOLEV–MORREY ESTIMATES FOR HYPOELLIPTIC OPERATORS WITH DRIFT ON HOMOGENEOUS GROUPS

Let G = ( RN,◦,δλ ) be a homogeneous group, Q be the homogeneous dimension of G, X0,X1,…,Xm be left invariant real vector fields on G and satisfy Hörmander’s rank condition on RN. Assume that X1,…,Xm (m ≤ N −1) are homogeneous of degree one and X0 is homogeneous of degree two with respect to the family of dilations ( δλ )λ>0. Consider the following hypoelliptic operator with drift on G (Formula Presented), where (aij) is a constant matrix satisfying the elliptic condition in Rm and a0 ≠ 0. In this paper, for this class of operators we obtain generalized weighted Sobolev-Morrey estimates by establishing boundedness of a large class of sublinear operators Tα, α ∈ [0,Q) generated by Calderón-Zygmund operators (α = 0) and generated by fractional integral operator (α > 0) on generalized weighted Morrey spaces and proving interpolation results in generalized weighted Sobolev-Morrey spaces on G © 2022. Journal of Mathematical Inequalities.All Rights Reserved

RIESZ TRANSFORMS ASSOCIATED WITH SCHRODINGER OPERATOR ON VANISHING GENERALIZED MORREY SPACES

In this paper, we study the boundedness of the dual Riesz transform R* and their commutators [b, R*] on generalized Morrey spaces M-p,phi(alpha,V) associated with Schrodinger operator and vanishing generalized Morrey spaces V M-p,phi(alpha,V) associated with Schrodinger operator. We find the sufficient conditions on the pair (phi(1) , phi(2)) which ensures the boundedness of the operator R* from one vanishing generalized Morrey space V M-p,phi 1(alpha,V) to another V M-p,phi 2(alpha,V)