Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces

Vagif Sabir Guliyev; Turhan Karaman; Rza Chingiz Mustafayev; Ayhan Şerbetçi

Czechoslovak Mathematical Journal (2014)

  • Volume: 64, Issue: 2, page 365-385
  • ISSN: 0011-4642

Abstract

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In this paper, the boundedness of a large class of sublinear commutator operators T b generated by a Calderón-Zygmund type operator on a generalized weighted Morrey spaces M p , ϕ ( w ) with the weight function w belonging to Muckenhoupt’s class A p is studied. When 1 < p < ∞ and b ∈ BMO , sufficient conditions on the pair ( ϕ 1 , ϕ 2 ) which ensure the boundedness of the operator T b from M p , ϕ 1 ( w ) to M p , ϕ 2 ( w ) are found. In all cases the conditions for the boundedness of T b are given in terms of Zygmund-type integral inequalities on ( ϕ 1 , ϕ 2 ) , which do not require any assumption on monotonicity of ϕ 1 ( x , r ) , ϕ 2 ( x , r ) in r . Then these results are applied to several particular operators such as the pseudo-differential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.

How to cite

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Guliyev, Vagif Sabir, et al. "Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces." Czechoslovak Mathematical Journal 64.2 (2014): 365-385. <http://eudml.org/doc/261997>.

@article{Guliyev2014,
abstract = {In this paper, the boundedness of a large class of sublinear commutator operators $T_\{b\}$ generated by a Calderón-Zygmund type operator on a generalized weighted Morrey spaces $M_\{p,\varphi \}(w)$ with the weight function $w$ belonging to Muckenhoupt’s class $A_\{p\}$ is studied. When $1<p<\infty $ and $b \in \{\rm BMO\}$, sufficient conditions on the pair $(\varphi _1,\varphi _2)$ which ensure the boundedness of the operator $T_\{b\}$ from $M_\{p,\varphi _1\}(w)$ to $M_\{p,\varphi _2\}(w)$ are found. In all cases the conditions for the boundedness of $T_\{b\}$ are given in terms of Zygmund-type integral inequalities on $(\varphi _1,\varphi _2)$, which do not require any assumption on monotonicity of $\varphi _1(x,r)$, $\varphi _2(x,r)$ in $r$. Then these results are applied to several particular operators such as the pseudo-differential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.},
author = {Guliyev, Vagif Sabir, Karaman, Turhan, Mustafayev, Rza Chingiz, Şerbetçi, Ayhan},
journal = {Czechoslovak Mathematical Journal},
keywords = {generalized weighted Morrey space; sublinear operator; commutator; BMO space; maximal operator; Calderón-Zygmund operator; generalized weighted Morrey space; sublinear operator; commutator; BMO; maximal operator; Calderón-Zygmund operator},
language = {eng},
number = {2},
pages = {365-385},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces},
url = {http://eudml.org/doc/261997},
volume = {64},
year = {2014},
}

TY - JOUR
AU - Guliyev, Vagif Sabir
AU - Karaman, Turhan
AU - Mustafayev, Rza Chingiz
AU - Şerbetçi, Ayhan
TI - Commutators of sublinear operators generated by Calderón-Zygmund operator on generalized weighted Morrey spaces
JO - Czechoslovak Mathematical Journal
PY - 2014
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 64
IS - 2
SP - 365
EP - 385
AB - In this paper, the boundedness of a large class of sublinear commutator operators $T_{b}$ generated by a Calderón-Zygmund type operator on a generalized weighted Morrey spaces $M_{p,\varphi }(w)$ with the weight function $w$ belonging to Muckenhoupt’s class $A_{p}$ is studied. When $1<p<\infty $ and $b \in {\rm BMO}$, sufficient conditions on the pair $(\varphi _1,\varphi _2)$ which ensure the boundedness of the operator $T_{b}$ from $M_{p,\varphi _1}(w)$ to $M_{p,\varphi _2}(w)$ are found. In all cases the conditions for the boundedness of $T_{b}$ are given in terms of Zygmund-type integral inequalities on $(\varphi _1,\varphi _2)$, which do not require any assumption on monotonicity of $\varphi _1(x,r)$, $\varphi _2(x,r)$ in $r$. Then these results are applied to several particular operators such as the pseudo-differential operators, Littlewood-Paley operator, Marcinkiewicz operator and Bochner-Riesz operator.
LA - eng
KW - generalized weighted Morrey space; sublinear operator; commutator; BMO space; maximal operator; Calderón-Zygmund operator; generalized weighted Morrey space; sublinear operator; commutator; BMO; maximal operator; Calderón-Zygmund operator
UR - http://eudml.org/doc/261997
ER -

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