Boundedness of a class of sublinear operators and their commutators on generalized Morrey spaces
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.
scientific article; zbMATH DE number 5946506
scientific article