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Zbl 1228.42017
Guliyev, Vagif S.; Aliyev, Seymur S.; Karaman, Turhan
Boundedness of a class of sublinear operators and their commutators on generalized Morrey spaces.
(English)
[J] Abstr. Appl. Anal. 2011, Article ID 356041, 18 p. (2011). ISSN 1085-3375; ISSN 1687-0409/e

Suppose that $T$ is a linear or a sublinear operator which satisfies for any $f\in L_1(\Bbb R^n)$ with compact support and $x\in (\operatorname{supp} f)^c$ $$|Tf(x)|\le c_0\int_{\Bbb R^n}\frac{|f(y)|}{|x-y|^n}\,dy,\tag1$$ 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 $f\in L_1(\Bbb R^n)$ with compact support and $x\in (\operatorname{supp} f)^c$ $$|Tf(x)|\le c_0\int_{\Bbb R^n}\frac{|f(y)||a(x)-a(y)|}{|x-y|^n}\,dy, \tag2$$ where $c_0$ is independent of $f$ and $x$. Let $\varphi(x,r)$ be a positive measurable function on $\Bbb R^n\times (0,\infty)$ and $1\le p<\infty$. We denote by $M_{p,\varphi}$ the generalized Morrey space of all functions $f\in L^{\text{loc}}_p(\Bbb R^n)$ with finite quasinorm $$\|f\|_{M_{p,\varphi}}=\sup_{x\in\Bbb R^n,\,r>0} \varphi(x,r)^{-1}|B(x,r)|^{-\frac{1}{p}}\|f\|_{L_p(B(x,r))}.$$ Also, by $WM_{p,\varphi}$ we denote the weak generalized Morrey space of all functions $f\in WL^{\text{loc}}_p(\Bbb R^n)$ for which $$\|f\|_{WM_{p,\varphi}}=\sup_{x\in\Bbb R^n,\,r>0} \varphi(x,r)^{-1}|B(x,r)|^{-\frac{1}{p}}\|f\|_{WL_p(B(x,r))}<\infty.$$ The authors prove the boundedness of the sublinear operator $T$ satisfying condition (1) generated by the CalderÃ³n-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.
[Yu Liu (Beijing)]
MSC 2000:
*42B20 Singular integrals, several variables
42B35 Function spaces arising in harmonic analysis

Keywords: CalderÃ³n-Zygmund operator; Morrey spaces; boundedness

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