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Boundedness of a class of sublinear operators and their commutators on generalized Morrey spaces. (English) Zbl 1228.42017
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 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.
Reviewer: Yu Liu (Beijing)

MSC:
42B20Singular and oscillatory integrals, several variables
42B35Function spaces arising in harmonic analysis
WorldCat.org
Full Text: DOI
References:
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