bounds for the commutators of singular integrals and maximal singular integrals with rough kernels
Authors:
Yanping Chen and Yong Ding
Journal:
Trans. Amer. Math. Soc. 367 (2015), 1585-1608
MSC (2010):
Primary 42B20, 42B25, 42B99
DOI:
https://doi.org/10.1090/S0002-9947-2014-06069-8
Published electronically:
July 29, 2014
MathSciNet review:
3286493
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: The commutator of convolution type Calderon-Zygmund singular integral operators with rough kernels are studied. The authors established the
boundedness of the commutators of singular integrals and maximal singular integrals with the kernel condition which is different from the condition
- [1] Hussain Al-Qassem and Ahmad Al-Salman, Rough Marcinkiewicz integral operators, Int. J. Math. Math. Sci. 27 (2001), no. 8, 495–503. MR 1869651, https://doi.org/10.1155/S0161171201006548
- [2] Ahmad Al-Salman and Yibiao Pan, Singular integrals with rough kernels, Canad. Math. Bull. 47 (2004), no. 1, 3–11. MR 2032262, https://doi.org/10.4153/CMB-2004-001-8
- [3] Josefina Álvarez, Richard J. Bagby, Douglas S. Kurtz, and Carlos Pérez, Weighted estimates for commutators of linear operators, Studia Math. 104 (1993), no. 2, 195–209. MR 1211818
- [4] Jean-Michel Bony, Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 2, 209–246 (French). MR 631751
- [5] Marco Bramanti and M. Cristina Cerutti, Commutators of singular integrals on homogeneous spaces, Boll. Un. Mat. Ital. B (7) 10 (1996), no. 4, 843–883 (English, with Italian summary). MR 1430157
- [6] A.-P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092–1099. MR 0177312
- [7] Calixto P. Calderón, On commutators of singular integrals, Studia Math. 53 (1975), no. 2, 139–174. MR 0380518
- [8] Yanping Chen and Yong Ding, 𝐿² boundedness for commutator of rough singular integral with variable kernel, Rev. Mat. Iberoam. 24 (2008), no. 2, 531–547. MR 2459202, https://doi.org/10.4171/RMI/545
- [9] Dong Xiang Chen and Shan Zhen Lu, 𝐿^{𝑝} boundedness of the parabolic Littlewood-Paley operator with rough kernel belonging to 𝐹_{𝛼}(𝑆ⁿ⁻¹), Acta Math. Sci. Ser. A Chin. Ed. 31 (2011), no. 2, 343–350 (Chinese, with English and Chinese summaries). MR 2828021
- [10] Jiecheng Chen, Dashan Fan, and Yibiao Pan, A note on a Marcinkiewicz integral operator, Math. Nachr. 227 (2001), 33–42. MR 1840553, https://doi.org/10.1002/1522-2616(200107)227:1<33::AID-MANA33>3.3.CO;2-S
- [11] Leslie C. Cheng and Yibiao Pan, 𝐿^{𝑝} bounds for singular integrals associated to surfaces of revolution, J. Math. Anal. Appl. 265 (2002), no. 1, 163–169. MR 1874263, https://doi.org/10.1006/jmaa.2001.7710
- [12] Jiecheng Chen and Chunjie Zhang, Boundedness of rough singular integral operators on the Triebel-Lizorkin spaces, J. Math. Anal. Appl. 337 (2008), no. 2, 1048–1052. MR 2386355, https://doi.org/10.1016/j.jmaa.2007.04.026
- [13] Filippo Chiarenza, Michele Frasca, and Placido Longo, Interior 𝑊^{2,𝑝} estimates for nondivergence elliptic equations with discontinuous coefficients, Ricerche Mat. 40 (1991), no. 1, 149–168. MR 1191890
- [14] Filippo Chiarenza, Michele Frasca, and Placido Longo, 𝑊^{2,𝑝}-solvability of the Dirichlet problem for nondivergence elliptic equations with VMO coefficients, Trans. Amer. Math. Soc. 336 (1993), no. 2, 841–853. MR 1088476, https://doi.org/10.1090/S0002-9947-1993-1088476-1
- [15] R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures Appl. (9) 72 (1993), no. 3, 247–286 (English, with English and French summaries). MR 1225511
- [16] R. R. Coifman, R. Rochberg, and Guido Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math. (2) 103 (1976), no. 3, 611–635. MR 0412721, https://doi.org/10.2307/1970954
- [17] Javier Duoandikoetxea and José L. Rubio de Francia, Maximal and singular integral operators via Fourier transform estimates, Invent. Math. 84 (1986), no. 3, 541–561. MR 837527, https://doi.org/10.1007/BF01388746
- [18] Javier Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc. 336 (1993), no. 2, 869–880. MR 1089418, https://doi.org/10.1090/S0002-9947-1993-1089418-5
- [19] Dashan Fan, Kanghui Guo, and Yibiao Pan, A note of a rough singular integral operator, Math. Inequal. Appl. 2 (1999), no. 1, 73–81. MR 1667793, https://doi.org/10.7153/mia-02-07
- [20] G. Di Fazio and M. A. Ragusa, Interior estimates in Morrey spaces for strong solutions to nondivergence form equations with discontinuous coefficients, J. Funct. Anal. 112 (1993), no. 2, 241–256. MR 1213138, https://doi.org/10.1006/jfan.1993.1032
- [21] Michael Frazier and Björn Jawerth, The 𝜙-transform and applications to distribution spaces, Function spaces and applications (Lund, 1986) Lecture Notes in Math., vol. 1302, Springer, Berlin, 1988, pp. 223–246. MR 942271, https://doi.org/10.1007/BFb0078877
- [22] Michael Frazier, Björn Jawerth, and Guido Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics, vol. 79, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1991. MR 1107300
- [23] J. García-Cuerva, E. Harboure, C. Segovia, and J. L. Torrea, Weighted norm inequalities for commutators of strongly singular integrals, Indiana Univ. Math. J. 40 (1991), no. 4, 1397–1420. MR 1142721, https://doi.org/10.1512/iumj.1991.40.40063
- [24] Loukas Grafakos, Classical and modern Fourier analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. MR 2449250
- [25] Loukas Grafakos and Atanas Stefanov, 𝐿^{𝑝} bounds for singular integrals and maximal singular integrals with rough kernels, Indiana Univ. Math. J. 47 (1998), no. 2, 455–469. MR 1647912, https://doi.org/10.1512/iumj.1998.47.1521
- [26] Loukas Grafakos, Petr Honzík, and Dmitry Ryabogin, On the 𝑝-independence boundedness property of Calderón-Zygmund theory, J. Reine Angew. Math. 602 (2007), 227–234. MR 2300457, https://doi.org/10.1515/CRELLE.2007.008
- [27] L. Greco and T. Iwaniec, New inequalities for the Jacobian, Ann. Inst. H. Poincaré Anal. Non Linéaire 11 (1994), no. 1, 17–35 (English, with English and French summaries). MR 1259100
- [28] Guoen Hu, 𝐿²(ℝⁿ) boundedness for the commutators of convolution operators, Nagoya Math. J. 163 (2001), 55–70. MR 1854388
- [29] Guoen Hu, 𝐿^{𝑝}(ℝⁿ) boundedness for the commutator of a homogeneous singular integral operator, Studia Math. 154 (2003), no. 1, 13–27. MR 1949046, https://doi.org/10.4064/sm154-1-2
- [30] Douglas S. Kurtz, Littlewood-Paley and multiplier theorems on weighted 𝐿^{𝑝} spaces, Trans. Amer. Math. Soc. 259 (1980), no. 1, 235–254. MR 561835, https://doi.org/10.1090/S0002-9947-1980-0561835-X
- [31] Richard Rochberg and Guido Weiss, Derivatives of analytic families of Banach spaces, Ann. of Math. (2) 118 (1983), no. 2, 315–347. MR 717826, https://doi.org/10.2307/2007031
- [32] E. M. Stein and G. Weiss, Interpolation of operators with change of measures, Trans. Amer. Math. Soc. 87 (1958), 159–172. MR 0092943, https://doi.org/10.1090/S0002-9947-1958-0092943-6
Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 42B20, 42B25, 42B99
Retrieve articles in all journals with MSC (2010): 42B20, 42B25, 42B99
Additional Information
Yanping Chen
Affiliation:
Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing 100083, The People’s Republic of China
Email:
yanpingch@126.com
Yong Ding
Affiliation:
School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems (BNU), Ministry of Education, Beijing 100875, The People’s Republic of China
Email:
dingy@bnu.edu.cn
DOI:
https://doi.org/10.1090/S0002-9947-2014-06069-8
Keywords:
Commutator,
singular integral,
maximal singular integral,
rough kernel,
BMO,
Bony paraproduct
Received by editor(s):
May 11, 2012
Received by editor(s) in revised form:
December 2, 2012
Published electronically:
July 29, 2014
Additional Notes:
The research was supported by NSF of China (Grant: 10901017, 11371057), NCET of China (Grant: NCET-11-0574), the Fundamental Research Funds for the Central Universities (FRF-TP-12-006B) and SRFDP of China (Grant: 20130003110003)
The first author is the corresponding author
Article copyright:
© Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.