Boundedness of a class of sublinear operators and their commutators on generalized Morrey spaces
type: Item
instance of: scholarly article
title: Boundedness of a class of sublinear operators and their commutators on generalized Morrey spaces
zbMATH Open document ID: 1228.42017
DOI: 10.1155/2011/356041
author: Vagif S. Guliyev
author: Seymur S. Aliev
author: Turhan Karaman
published in: Abstract and Applied Analysis
publication date: 2011-09-09T00:00:00Z
review text: Suppose that \(T\) is a linear or a sublinear operator which satisfies for any \(f\in L_1(\mathbb R^n)\) with compact support and \(x\in (\operatorname{supp} f)^c\) \[ |Tf(x)|\leq c_0\int_{\mathbb R^n}\frac{|f(y)|}{|x-y|^n}\,dy,\tag{1} \] where \(c_0\) is independent of \(f\) and \(x\). For a function \(a\), suppose that \(T_a\) is a commutator generated by \(T\) and \(a\) satisfies for any \(f\in L_1(\mathbb R^n)\) with compact support and \(x\in (\operatorname{supp} f)^c\) \[ |Tf(x)|\leq c_0\int_{\mathbb R^n}\frac{|f(y)||a(x)-a(y)|}{|x-y|^n}\,dy, \tag{2} \] where \(c_0\) is independent of \(f\) and \(x\). Let \(\varphi(x,r)\) be a positive measurable function on \(\mathbb R^n\times (0,\infty)\) and \(1\leq p<\infty\). We denote by \(M_{p,\varphi}\) the generalized Morrey space of all functions \(f\in L^{\text{loc}}_p(\mathbb R^n)\) with finite quasinorm \[ \|f\|_{M_{p,\varphi}}=\sup_{x\in\mathbb R^n,\,r>0} \varphi(x,r)^{-1}|B(x,r)|^{-\frac{1}{p}}\|f\|_{L_p(B(x,r))}. \] Also, by \(WM_{p,\varphi}\) we denote the weak generalized Morrey space of all functions \(f\in WL^{\text{loc}}_p(\mathbb R^n)\) for which \[ \|f\|_{WM_{p,\varphi}}=\sup_{x\in\mathbb R^n,\,r>0} \varphi(x,r)^{-1}|B(x,r)|^{-\frac{1}{p}}\|f\|_{WL_p(B(x,r))}<\infty. \] The authors prove the boundedness of the sublinear operator \(T\) satisfying condition (1) generated by the Calderón-Zygmund operator from one generalized Morrey space \(M_{p,\varphi_1}\) to another space \(M_{p,\varphi_2}\) for \(1<p<\infty\) and from \(M_{1,\varphi_1}\) to the weak space \(WM_{1,\varphi_2}\). When \(a\in \text{BMO}\), they find a sufficient condition on the pair \((\varphi_1,\varphi_2)\) which ensures the boundedness of the commutator \(T_a\) from \(M_{p,\varphi_1}\) to \(M_{p,\varphi_2}\) for \(1<p<\infty\). Finally, they apply their results to several particular operators such as pseudodifferential operators, the Littlewood-Paley operator, the Marcinkiewicz operator, and the Bochner-Riesz operator.
Mathematics Subject Classification ID: 42B20
Mathematics Subject Classification ID: 42B35
zbMATH DE Number: 5946506
zbMATH Keywords: Calderón-Zygmund operator
zbMATH Keywords: Morrey spaces
zbMATH Keywords: boundedness
reviewed by: Yu Liu
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2011/356041
OpenAlex ID: W2144078873
Wikidata QID: Q58654154
cites work: Q3941791
cites work: Q3700300
cites work: Q3446088
cites work: Q5607484
cites work: Q3142876
cites work: Q3756738
cites work: Q4301002
cites work: Singular integrals and commutators on homogeneous groups
cites work: On the Solutions of Quasi-Linear Elliptic Partial Differential Equations
cites work: On the theory of \({\mathcal L}_{p, \lambda}\) spaces
cites work: Q5748393
cites work: Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients
cites work: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces
cites work: Boundedness of the maximal, potential and singular operators in the generalized variable exponent Morrey spaces
cites work: Strongly singular Calderón-Zygmund operator and commutator on Morrey type spaces
cites work: Q4694410
cites work: Hardy-Littlewood Maximal Operator, Singular Integral Operators and the Riesz Potentials on Generalized Morrey Spaces
cites work: Necessary and sufficient conditions for the boundedness of the fractional maximal operator in local Morrey-type spaces
cites work: Necessary and sufficient conditions for the boundedness of fractional maximal operators in local Morrey-type spaces
cites work: Q3081805
cites work: Boundedness of the fractional maximal operator in local Morrey-type spaces
cites work: Q4484818
cites work: Q4236453
cites work: Sublinear operators with rough kernel on generalized Morrey spaces
cites work: On embeddings between classical Lorentz spaces
cites work: On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces.
cites work: Factorization theorems for Hardy spaces in several variables
cites work: Interior \(W^{2,p}\) estimates for non-divergence elliptic equations with discontinuous coefficients
cites work: W 2,p -Solvability of the Dirichlet Problem for Nondivergence Elliptic Equations with VMO Coefficients
cites work: Pseudodifferential operators and nonlinear PDE
cites work: Weighted Sobolev Spaces and Pseudodifferential Operators with Smooth Symbols
cites work: On the Functions of Littlewood-Paley, Lusin, and Marcinkiewicz
cites work: Q4461729
cites work: A note on the Marcinkiewicz integral
cites work: The boundedness of maximal Bochner-Riesz operator and maximal commutator on Morrey type spaces
cites work: Weighted weak type inequalities for maximal commutators of Bochner-Riesz operator
description: scientific article; zbMATH DE number 5946506