Acknowledgement
Supported by : NNSF of China
References
- S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen inequalities, Trans. Amer. Math. Soc. 340 (1993), no. 1, 253-272. https://doi.org/10.1090/S0002-9947-1993-1124164-0
- A. Bui, J. Conde-Alonso, X. T. Duong, and M. Hormozi, Weighted bounds for multilinear operators with non-smooth kernels, arxiv:1506.07752.
- A.-P. Calderon, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092-1099. https://doi.org/10.1073/pnas.53.5.1092
- C. P. Calderon, On commutators of singular integrals, Studia Math. 53 (1975), no. 2, 139-174. https://doi.org/10.4064/sm-53-2-139-174
- D. Chung, M. C. Pereyra, and C. Perez, Sharp bounds for general commutators on weighted Lebesgue spaces, Trans. Amer. Math. Soc. 364 (2012), no. 3, 1163-1177. https://doi.org/10.1090/S0002-9947-2011-05534-0
- R. R. Coifman and Y. Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315-331. https://doi.org/10.1090/S0002-9947-1975-0380244-8
- R. R. Coifman and Y. Meyer, Au dela des operateurs pseudo-differentiels, Asterisque, 57, Societe Mathematique de France, Paris, 1978.
- J. M. Conde-Alonso and G. Rey, A pointwise estimate for positive dyadic shifts and some applications, Math. Ann. 365 (2016), no. 3-4, 1111-1135. https://doi.org/10.1007/s00208-015-1320-y
- D. Cruz-Uribe, J. M. Martell, and C. Perez, Sharp weighted estimates for classical operators, Adv. Math. 229 (2012), no. 1, 408-441. https://doi.org/10.1016/j.aim.2011.08.013
- W. Damian, M. Hormozi, and K. Li, New bounds for bilinear Calderon-Zygmund operators and applications, arxiv:1512.02400.
- O. Dragicevic, L. Grafakos, M. Pereyra, and S. Petermichl, Extrapolation and sharp norm estimates for classical operators on weighted Lebesgue spaces, Publ. Mat. 49 (2005), 73-91. https://doi.org/10.5565/PUBLMAT_49105_03
- X. Duong, R. Gong, L. Grafakos, J. Li, and L. Yan,, Maximal operator for multilinear singular integrals with non-smooth kernels, Indiana Univ. Math. J. 58 (2009), no. 6, 2517-2541. https://doi.org/10.1512/iumj.2009.58.3803
- X. T. Duong, L. Grafakos, and L. Yan, Multilinear operators with non-smooth kernels and commutators of singular integrals, Trans. Amer. Math. Soc. 362 (2010), no. 4, 2089-2113. https://doi.org/10.1090/S0002-9947-09-04867-3
- C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. https://doi.org/10.2307/2373450
- L. Grafakos, Modern Fourier Analysis, second edition, Graduate Texts in Mathematics, 250, Springer, New York, 2009.
- L. Grafakos, L. Liu, and D. Yang, Multiple-weighted norm inequalities for maximal multi-linear singular integrals with non-smooth kernels, Proc. Roy. Soc. Edinburgh Sect. A 141 (2011), no. 4, 755-775. https://doi.org/10.1017/S0308210509001383
- L. Grafakos and R. H. Torres, Multilinear Calderon-Zygmund theory, Adv. Math. 165 (2002), no. 1, 124-164. https://doi.org/10.1006/aima.2001.2028
- G. Hu and D. Li, A Cotlar type inequality for the multilinear singular integral operators and its applications, J. Math. Anal. Appl. 290 (2004), no. 2, 639-653. https://doi.org/10.1016/j.jmaa.2003.10.037
- G. Hu and K. Li, Weighted vector-valued inequalities for a class of multilinear singular integral operators, Proc. Edinb. Math. Soc., to appear.
- G. E. Hu and Y. P. Zhu, Weighted norm inequalities with general weights for the commutator of Calderon, Acta Math. Sin. (Engl. Ser.) 29 (2013), no. 3, 505-514. https://doi.org/10.1007/s10114-012-1352-0
- T. P. Hytonen, The sharp weighted bound for general Calderon-Zygmund operators, Ann. of Math. (2) 175 (2012), no. 3, 1473-1506. https://doi.org/10.4007/annals.2012.175.3.9
- T. P. Hytonen, M. T. Lacey, and C. Perez, Sharp weighted bounds for the q-variation of singular integrals, Bull. Lond. Math. Soc. 45 (2013), no. 3, 529-540. https://doi.org/10.1112/blms/bds114
-
T. Hytonen and C. Perez, Sharp weighted bounds involving
$A_{\infty}$ , Anal. PDE 6 (2013), no. 4, 777-818. https://doi.org/10.2140/apde.2013.6.777 - A. K. Lerner, On pointwise estimates involving sparse operators, New York J. Math. 22 (2016), 341-349.
- K. Lerner and F. Nazarov, Intuitive dyadic calculus: the basics, arXiv:1508.05639.
- A. Lerner, S. Ombrossi, C. Perez, R. H. Torres, and R. Trojillo-Gonzalez, New maximal functions and multiple weights for the multilinear Calderon-Zygmund theory, Adv. Math. 220 (2009), no. 4, 1222-1264. https://doi.org/10.1016/j.aim.2008.10.014
- A. K. Lerner, S. Ombrosi, and I. P. Rivera-Rios, On pointwise and weighted estimates for commutators of Calderon-Zygmund operators, Adv. Math. 319 (2017), 153-181. https://doi.org/10.1016/j.aim.2017.08.022
-
K. Li, Sparse domination theorem for mltilinear singular integral operators with
$L^r$ -Hormander condition, arxiv:1606.03952. https://doi.org/10.1103/PhysRevD.95.124034 - K. Li, K. Moen, and W. Sun, The sharp weighted bound for multilinear maximal functions and Calderon-Zygmund operators, J. Fourier Anal. Appl. 20 (2014), no. 4, 751-765. https://doi.org/10.1007/s00041-014-9326-5
- K. Li and W. Sun, Weak and strong type weighted estimates for multilinear Calderon-Zygmund operators, Adv. Math. 254 (2014), 736-771. https://doi.org/10.1016/j.aim.2013.12.027
- B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. https://doi.org/10.1090/S0002-9947-1972-0293384-6
- C. Perez, G. Pradolini, R. H. Torres, and R. Trujillo-Gonzalez, End-point estimates for iterated commutators of multilinear singular integrals, Bull. Lond. Math. Soc. 46 (2014), no. 1, 26-42. https://doi.org/10.1112/blms/bdt065
-
S. Petermichl, The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical
$A_p$ characteristic, Amer. J. Math. 129 (2007), no. 5, 1355-1375. https://doi.org/10.1353/ajm.2007.0036 - S. Petermichl, The sharp weighted bound for the Riesz transforms, Proc. Amer. Math. Soc. 136 (2008), no. 4, 1237-1249. https://doi.org/10.1090/S0002-9939-07-08934-4
- M. M. Rao and Z. D. Ren, Theory of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 146, Marcel Dekker, Inc., New York, 1991.
- E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993.