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Higher order commutators for a class of rough operators. (English) Zbl 1038.42013
For a BMO$(\nu)$ function $b(x)$, the higher order commutators generated by the fractional integral operator and the maximal operator are defined by $$T_{\Omega,\alpha,b}^{m}f(x) = \int_{\Bbb{R}^{n}} \frac{\Omega(x-y)}{\vert x-y\vert ^{n-\alpha}} [b(x) - b(y)]^{m} \, f(y) \, dy$$ and $$M_{\Omega,\alpha,b}^{m}f(x) = \sup_{r > 0} \frac{1}{r^{n-\alpha}}\int_{\vert x-y\vert <r} \vert \Omega(x-y)\vert [b(x) - b(y)]^{m} \, \vert f(y)\vert \, dy.$$ Here a function $b(x)$ is said to belong to BMO$(\nu)$ if there is a constant $C > 0$ such that for any $Q$ in $\Bbb{R}^{n}$ with its sides parallel to the coordinate axes and for a nonnegative locally integrable function $\nu$ on $\Bbb{R}^{n}$ $\int_{Q} \vert b(x)-b_{Q}\vert \, dx \leq C \, \int_{Q} \nu(x) \, dx$, where $b_{Q} = (1/\vert Q\vert )\int_{Q}b(x) \, dx$. The authors prove the $(L^{p}(u^p), L^{q}(v^q))$-boundedness of the higher order commutators $T_{\Omega,\alpha,b}^{m}$ and $M_{\Omega,\alpha,b}^{m}$. Their results improve the known result due to {\it C. Segovia} and {\it 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:
 42B20 Singular and oscillatory integrals, several variables 42B25 Maximal functions, Littlewood-Paley theory
Full Text:
##### References:
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