## Commutators with Lipschitz functions and nonintegral operators.(English)Zbl 1271.35043

Summary: Let $$T$$ be a singular nonintegral operator; that is, it does not have an integral representation by a kernel with size estimates, even rough. In this paper, we consider the boundedness of commutators with $$T$$ and Lipschitz functions. Applications include spectral multipliers of self-adjoint, positive operators, Riesz transforms of second-order divergence form operators, and fractional power of elliptic operators.

### MSC:

 35J75 Singular elliptic equations 45E99 Singular integral equations
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### References:

 [1] R. R. Coifman, R. Rochberg, and G. Weiss, “Factorization theorems for Hardy spaces in several variables,” Annals of Mathematics, vol. 103, no. 3, pp. 611-635, 1976. · Zbl 0326.32011 [2] S. Janson, “Mean oscillation and commutators of singular integral operators,” Arkiv för Matematik, vol. 16, no. 2, pp. 263-270, 1978. · Zbl 0404.42013 [3] M. Bramanti and M. C. Cerutti, “Commutators of singular integrals on homogeneous spaces,” Bollettino dell’Unione Matematica Italiana, vol. 10, no. 4, pp. 843-883, 1996. · Zbl 0913.42013 [4] X. T. Duong and L. X. Yan, “Commutators of BMO functions and singular integral operators with non-smooth kernels,” Bulletin of the Australian Mathematical Society, vol. 67, no. 2, pp. 187-200, 2003. · Zbl 1023.42010 [5] X. T. Duong and A. MacIntosh, “Singular integral operators with non-smooth kernels on irregular domains,” Revista Matemática Iberoamericana, vol. 15, no. 2, pp. 233-265, 1999. · Zbl 0980.42007 [6] G. Hu and D. Yang, “Maximal commutators of BMO functions and singular integral operators with non-smooth kernels on spaces of homogeneous type,” Journal of Mathematical Analysis and Applications, vol. 354, no. 1, pp. 249-262, 2009. · Zbl 1170.42009 [7] P. Auscher and J. M. Martell, “Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part I: general operator theory and weights,” Advances in Mathematics, vol. 212, no. 1, pp. 225-276, 2007. · Zbl 1213.42030 [8] S. Blunck and P. C. Kunstmann, “Calderón-Zygmund theory for non-integral operators and the H\infty functional calculus,” Revista Matemática Iberoamericana, vol. 19, no. 3, pp. 919-942, 2003. · Zbl 1057.42010 [9] P. Auscher and J. M. Martell, “Weighted norm inequalities for fractional operators,” Indiana University Mathematics Journal, vol. 57, no. 4, pp. 1845-1869, 2008. · Zbl 1158.42010 [10] M. Paluszyński, “Characterization of the Besov spaces via the commutator operator of Coifman, Rochberg and Weiss,” Indiana University Mathematics Journal, vol. 44, no. 1, pp. 1-18, 1995. · Zbl 0838.42006 [11] S. Chanillo, “A note on commutators,” Indiana University Mathematics Journal, vol. 31, no. 1, pp. 7-16, 1982. · Zbl 0523.42015 [12] A. Gogatishvili and V. Kokilashvili, “Criteria of strong type two-weighted inequalities for fractional maximal functions,” Georgian Mathematical Journal, vol. 3, no. 5, pp. 423-446, 1996. · Zbl 0884.42015 [13] X. T. Duong and L. X. Yan, “Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates,” Journal of the Mathematical Society of Japan, vol. 63, no. 1, pp. 295-319, 2011. · Zbl 1221.42024 [14] P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, and Ph. Tchamitchian, “The solution of the Kato square root problem for second order elliptic operators on \Bbb Rn,” Annals of Mathematics, vol. 156, no. 2, pp. 633-654, 2002. · Zbl 1128.35316 [15] P. Auscher and J. M. Martell, “Weighted norm inequalities, off-diagonal estimates and elliptic operators. Part III: harmonic analysis of elliptic operators,” Journal of Functional Analysis, vol. 241, no. 2, pp. 703-746, 2006. · Zbl 1213.42029
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