𝐿^{𝑝} bounds for the commutators of singular integrals and maximal singular integrals with rough kernels

Chen, Yanping; Ding, Yong

doi:10.1090/S0002-9947-2014-06069-8

$L^p$ bounds for the commutators of singular integrals and maximal singular integrals with rough kernels
HTML articles powered by AMS MathViewer

by Yanping Chen and Yong Ding PDF

Trans. Amer. Math. Soc. 367 (2015), 1585-1608 Request permission

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