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Higher order commutators for a class of rough operators. (English) Zbl 1038.42013

For a BMO(ν) function b(x), the higher order commutators generated by the fractional integral operator and the maximal operator are defined by

T Ω,α,b m f(x)= n Ω(x-y) |x-y| n-α [b(x)-b(y)] m f(y)dy

and

M Ω,α,b m f(x)=sup r>0 1 r n-α |x-y|<r |Ω(x-y)|[b(x)-b(y)] m |f(y)|dy·

Here a function b(x) is said to belong to BMO(ν) if there is a constant C>0 such that for any Q in n with its sides parallel to the coordinate axes and for a nonnegative locally integrable function ν on n Q |b(x)-b Q |dxC Q ν(x)dx, where b Q =(1/|Q|) Q b(x)dx. The authors prove the (L p (u p ),L q (v q ))-boundedness of the higher order commutators T Ω,α,b m and M Ω,α,b m . Their results improve the known result due to C. Segovia and J. L. Torrea [Trans. Am. Math. Soc. 336, No. 2, 537–556 (1993; Zbl 0799.42009)] in the sense that the cancellation condition and the smoothness condition have been removed.

MSC:
42B20Singular and oscillatory integrals, several variables
42B25Maximal functions, Littlewood-Paley theory
References:
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[9]Segovia, C. andTorrea, J. L., Higher order commutators for vector-valued Calderón-Zygmund operators,Trans. Amer. Math. Soc. 336 (1993), 537–556. · Zbl 0799.42009 · doi:10.2307/2154362
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