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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.

42B20Singular and oscillatory integrals, several variables
42B25Maximal functions, Littlewood-Paley theory
Full Text: DOI
[1] Chanillo, S., Watson, D. andWheeden, R. L., Some integral and maximal operators related to star like sets,Studia Math. 107 (1993), 223--255. · Zbl 0809.42008
[2] Ding, Y., Weighted boundedness for commutators of integral operators of fractionnal order with rough kernels,Beijing Shifan Daxue Xuebao 32 (1996), 157--161 (Chinese). · Zbl 0896.42012
[3] Ding, Y., Weighted boundedness for commutators of a class of rough maximal operators,Kexue Tongbao (Chinese) 41 (1996), 385--388 (Chinese).
[4] Ding, Y., Weak type bounds for a class of rough operators with power weights,Proc. Amer. Math. Soc. 125 (1997), 2939--2942. · Zbl 0887.42008 · doi:10.1090/S0002-9939-97-03914-2
[5] Ding, Y. andLu, S. Z., Weighted norm inequalities for fractional integral operators with rough kernel,Canad. J. Math. 50 (1998), 29--39. · Zbl 0905.42010 · doi:10.4153/CJM-1998-003-1
[6] García-Cuerva, J. andRubio de Francia, J. L.,Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985. · Zbl 0578.46046
[7] Hedberg, L. I., On certain convolution inequalities,Proc. Amer. Math. Soc. 36 (1972), 505--510. · Zbl 0283.26003 · doi:10.1090/S0002-9939-1972-0312232-4
[8] Muckenhoupt, B. andWheeden, R. L., Weighted norm inequalities for singular and fractional integrals,Trans. Amer. Math. Soc. 161 (1971), 249--258. · Zbl 0226.44007 · doi:10.1090/S0002-9947-1971-0285938-7
[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
[10] Welland, G. V., Weighted norm inequalities for fractional integrals,Proc. Amer. Math. Soc. 51 (1975), 143--148. · Zbl 0306.26007 · doi:10.1090/S0002-9939-1975-0369641-X