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

For a BMO$\left(\nu \right)$ function $b\left(x\right)$, the higher order commutators generated by the fractional integral operator and the maximal operator are defined by

${T}_{{\Omega },\alpha ,b}^{m}f\left(x\right)={\int }_{{ℝ}^{n}}\frac{{\Omega }\left(x-y\right)}{{|x-y|}^{n-\alpha }}{\left[b\left(x\right)-b\left(y\right)\right]}^{m}\phantom{\rule{0.166667em}{0ex}}f\left(y\right)\phantom{\rule{0.166667em}{0ex}}dy$

and

${M}_{{\Omega },\alpha ,b}^{m}f\left(x\right)=\underset{r>0}{sup}\frac{1}{{r}^{n-\alpha }}{\int }_{|x-y|

Here a function $b\left(x\right)$ is said to belong to BMO$\left(\nu \right)$ if there is a constant $C>0$ such that for any $Q$ in ${ℝ}^{n}$ with its sides parallel to the coordinate axes and for a nonnegative locally integrable function $\nu$ on ${ℝ}^{n}$ ${\int }_{Q}|b\left(x\right)-{b}_{Q}|\phantom{\rule{0.166667em}{0ex}}dx\le C\phantom{\rule{0.166667em}{0ex}}{\int }_{Q}\nu \left(x\right)\phantom{\rule{0.166667em}{0ex}}dx$, where ${b}_{Q}=\left(1/|Q|\right){\int }_{Q}b\left(x\right)\phantom{\rule{0.166667em}{0ex}}dx$. The authors prove the $\left({L}^{p}\left({u}^{p}\right),{L}^{q}\left({v}^{q}\right)\right)$-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 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:
 42B20 Singular and oscillatory integrals, several variables 42B25 Maximal functions, Littlewood-Paley theory
##### References:
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