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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 fL 1 ( n ) with compact support and x(suppf) c

|Tf(x)|c 0 n |f(y)| |x-y| n dy,(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 fL 1 ( n ) with compact support and x(suppf) c

|Tf(x)|c 0 n |f(y)||a(x)-a(y)| |x-y| n dy,(2)

where c 0 is independent of f and x.

Let ϕ(x,r) be a positive measurable function on n ×(0,) and 1p<. We denote by M p,ϕ the generalized Morrey space of all functions fL p loc ( n ) with finite quasinorm

f M p,ϕ =sup x n ,r>0 ϕ(x,r) -1 |B(x,r)| -1 p f L p (B(x,r)) ·

Also, by WM p,ϕ we denote the weak generalized Morrey space of all functions fWL p loc ( n ) for which

f WM p,ϕ =sup x n ,r>0 ϕ(x,r) -1 |B(x,r)| -1 p f WL p (B(x,r)) <·

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,ϕ 1 to another space M p,ϕ 2 for 1<p< and from M 1,ϕ 1 to the weak space WM 1,ϕ 2 . When aBMO, they find a sufficient condition on the pair (ϕ 1 ,ϕ 2 ) which ensures the boundedness of the commutator T a from M p,ϕ 1 to M p,ϕ 2 for 1<p<. 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