Multilinear operators with non-smooth kernels and commutators of singular integrals
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- by Xuan Thinh Duong, Loukas Grafakos and Lixin Yan PDF
- Trans. Amer. Math. Soc. 362 (2010), 2089-2113 Request permission
Abstract:
We obtain endpoint estimates for multilinear singular integral operators whose kernels satisfy regularity conditions significantly weaker than those of the standard Calderón-Zygmund kernels. As a consequence, we deduce endpoint $L^1 \times \dots \times L^1$ to weak $L^{1/m}$ estimates for the $m$th-order commutator of Calderón. Our results reproduce known estimates for $m = 1, 2$ but are new for $m \ge 3$. We also explore connections between the $2$nd-order higher-dimensional commutator and the bilinear Hilbert transform and deduce some new off-diagonal estimates for the former.References
- 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
- Michael Christ and Jean-Lin Journé, Polynomial growth estimates for multilinear singular integral operators, Acta Math. 159 (1987), no. 1-2, 51–80. MR 906525, DOI 10.1007/BF02392554
- R. R. Coifman and Yves Meyer, On commutators of singular integrals and bilinear singular integrals, Trans. Amer. Math. Soc. 212 (1975), 315–331. MR 380244, DOI 10.1090/S0002-9947-1975-0380244-8
- Ronald R. Coifman and Yves Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque, vol. 57, Société Mathématique de France, Paris, 1978 (French). With an English summary. MR 518170
- Yves Meyer and R. R. Coifman, Ondelettes et opérateurs. III, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1991 (French). Opérateurs multilinéaires. [Multilinear operators]. MR 1160989
- Xuan Thinh Duong and Alan MacIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana 15 (1999), no. 2, 233–265. MR 1715407, DOI 10.4171/RMI/255
- Loukas Grafakos, Classical and modern Fourier analysis, Pearson Education, Inc., Upper Saddle River, NJ, 2004. MR 2449250
- Loukas Grafakos and Nigel Kalton, Some remarks on multilinear maps and interpolation, Math. Ann. 319 (2001), no. 1, 151–180. MR 1812822, DOI 10.1007/PL00004426
- Loukas Grafakos and Xiaochun Li, Uniform bounds for the bilinear Hilbert transforms. I, Ann. of Math. (2) 159 (2004), no. 3, 889–933. MR 2113017, DOI 10.4007/annals.2004.159.889
- Loukas Grafakos and Rodolfo H. Torres, On multilinear singular integrals of Calderón-Zygmund type, Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations (El Escorial, 2000), 2002, pp. 57–91. MR 1964816, DOI 10.5565/PUBLMAT_{E}sco02_{0}4
- Loukas Grafakos and Rodolfo H. Torres, Multilinear Calderón-Zygmund theory, Adv. Math. 165 (2002), no. 1, 124–164. MR 1880324, DOI 10.1006/aima.2001.2028
- C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107–115. MR 284802, DOI 10.2307/2373450
- Carlos E. Kenig and Elias M. Stein, Multilinear estimates and fractional integration, Math. Res. Lett. 6 (1999), no. 1, 1–15. MR 1682725, DOI 10.4310/MRL.1999.v6.n1.a1
- Michael Lacey and Christoph Thiele, $L^p$ estimates on the bilinear Hilbert transform for $2<p<\infty$, Ann. of Math. (2) 146 (1997), no. 3, 693–724. MR 1491450, DOI 10.2307/2952458
- Michael Lacey and Christoph Thiele, On Calderón’s conjecture, Ann. of Math. (2) 149 (1999), no. 2, 475–496. MR 1689336, DOI 10.2307/120971
- Xiaochun Li, Uniform bounds for the bilinear Hilbert transforms. II, Rev. Mat. Iberoam. 22 (2006), no. 3, 1069–1126. MR 2320411, DOI 10.4171/RMI/483
- Takafumi Murai, A real variable method for the Cauchy transform, and analytic capacity, Lecture Notes in Mathematics, vol. 1307, Springer-Verlag, Berlin, 1988. MR 944308, DOI 10.1007/BFb0078078
- Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy; Monographs in Harmonic Analysis, III. MR 1232192
Additional Information
- Xuan Thinh Duong
- Affiliation: Department of Mathematics, Macquarie University, NSW, 2109, Australia
- MR Author ID: 271083
- Email: duong@ics.mq.edu.au
- Loukas Grafakos
- Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
- MR Author ID: 288678
- ORCID: 0000-0001-7094-9201
- Email: loukas@math.missouri.edu
- Lixin Yan
- Affiliation: Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
- MR Author ID: 618148
- Email: mcsylx@mail.sysu.edu.cn
- Received by editor(s): January 28, 2008
- Received by editor(s) in revised form: May 9, 2008
- Published electronically: October 20, 2009
- Additional Notes: The first author was supported by a grant from the Australia Research Council.
The second author was supported by grant DMS $0400387$ of the US National Science Foundation and by the University of Missouri Research Council
The third author was supported by NCET of Ministry of Education of China and NNSF of China (Grant No. 10771221). - © Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc. 362 (2010), 2089-2113
- MSC (2000): Primary 42B20, 42B25; Secondary 46B70, 47G30
- DOI: https://doi.org/10.1090/S0002-9947-09-04867-3
- MathSciNet review: 2574888