$L^p$ bounds for the commutators of singular integrals and maximal singular integrals with rough kernels
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Abstract:
The commutator of convolution type Calderon-Zygmund singular integral operators with rough kernels $p.v. \frac {\Omega (x)}{|x|^n}$ are studied. The authors established the $L^p (1<p<\infty )$ boundedness of the commutators of singular integrals and maximal singular integrals with the kernel condition which is different from the condition $\Omega \in H^1(S^{n-1}).$References
- Hussain Al-Qassem and Ahmad Al-Salman, Rough Marcinkiewicz integral operators, Int. J. Math. Math. Sci. 27 (2001), no. 8, 495–503. MR 1869651, DOI 10.1155/S0161171201006548
- Ahmad Al-Salman and Yibiao Pan, Singular integrals with rough kernels, Canad. Math. Bull. 47 (2004), no. 1, 3–11. MR 2032262, DOI 10.4153/CMB-2004-001-8
- 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, DOI 10.4064/sm-104-2-195-209
- 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
- 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
- A.-P. Calderón, Commutators of singular integral operators, Proc. Nat. Acad. Sci. U.S.A. 53 (1965), 1092–1099. MR 177312, DOI 10.1073/pnas.53.5.1092
- Calixto P. Calderón, On commutators of singular integrals, Studia Math. 53 (1975), no. 2, 139–174. MR 380518, DOI 10.4064/sm-53-2-139-174
- Yanping Chen and Yong Ding, $L^2$ boundedness for commutator of rough singular integral with variable kernel, Rev. Mat. Iberoam. 24 (2008), no. 2, 531–547. MR 2459202, DOI 10.4171/RMI/545
- Dong Xiang Chen and Shan Zhen Lu, $L^p$ boundedness of the parabolic Littlewood-Paley operator with rough kernel belonging to $F_\alpha (S^{n-1})$, Acta Math. Sci. Ser. A (Chinese Ed.) 31 (2011), no. 2, 343–350 (Chinese, with English and Chinese summaries). MR 2828021
- Jiecheng Chen, Dashan Fan, and Yibiao Pan, A note on a Marcinkiewicz integral operator, Math. Nachr. 227 (2001), 33–42. MR 1840553, DOI 10.1002/1522-2616(200107)227:1<33::AID-MANA33>3.3.CO;2-S
- 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, DOI 10.1006/jmaa.2001.7710
- 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, DOI 10.1016/j.jmaa.2007.04.026
- 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
- 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, DOI 10.1090/S0002-9947-1993-1088476-1
- 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
- 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 412721, DOI 10.2307/1970954
- 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, DOI 10.1007/BF01388746
- Javier Duoandikoetxea, Weighted norm inequalities for homogeneous singular integrals, Trans. Amer. Math. Soc. 336 (1993), no. 2, 869–880. MR 1089418, DOI 10.1090/S0002-9947-1993-1089418-5
- 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, DOI 10.7153/mia-02-07
- 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, DOI 10.1006/jfan.1993.1032
- Michael Frazier and Björn Jawerth, The $\phi$-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, DOI 10.1007/BFb0078877
- 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, DOI 10.1090/cbms/079
- 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, DOI 10.1512/iumj.1991.40.40063
- Loukas Grafakos, Classical and modern Fourier analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. MR 2449250
- 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, DOI 10.1512/iumj.1998.47.1521
- 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, DOI 10.1515/CRELLE.2007.008
- L. Greco and T. Iwaniec, New inequalities for the Jacobian, Ann. Inst. H. Poincaré C Anal. Non Linéaire 11 (1994), no. 1, 17–35 (English, with English and French summaries). MR 1259100, DOI 10.1016/S0294-1449(16)30194-9
- Guoen Hu, $L^2({\Bbb R}^n)$ boundedness for the commutators of convolution operators, Nagoya Math. J. 163 (2001), 55–70. MR 1854388, DOI 10.1017/S002776300000790X
- Guoen Hu, $L^p(\Bbb R^n)$ boundedness for the commutator of a homogeneous singular integral operator, Studia Math. 154 (2003), no. 1, 13–27. MR 1949046, DOI 10.4064/sm154-1-2
- 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, DOI 10.1090/S0002-9947-1980-0561835-X
- Richard Rochberg and Guido Weiss, Derivatives of analytic families of Banach spaces, Ann. of Math. (2) 118 (1983), no. 2, 315–347. MR 717826, DOI 10.2307/2007031
- E. M. Stein and G. Weiss, Interpolation of operators with change of measures, Trans. Amer. Math. Soc. 87 (1958), 159–172. MR 92943, DOI 10.1090/S0002-9947-1958-0092943-6
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
- MR Author ID: 213750
- Email: dingy@bnu.edu.cn
- 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 - © Copyright 2014
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - 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
- MathSciNet review: 3286493