Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Multilinear operators with non-smooth kernels and commutators of singular integrals


Authors: Xuan Thinh Duong, Loukas Grafakos and Lixin Yan
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
Published electronically: October 20, 2009
MathSciNet review: 2574888
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [AC] A.P. Calderón, Commutators of singular integrals, Proc. Nat. Acad. Sci. U.S.A., 53 (1965), 1092-1099. MR 0177312 (31:1575)
  • [CC] C.P. Calderón, On commutators of singular integrals, Studia Math., 53 (1975), 139-174. MR 0380518 (52:1418)
  • [CJ] M. Christ and J.-L. Journé, Polynomial growth estimates for multilinear singular integral operators, Acta Math., 159 (1987), 51-80. MR 906525 (89a:42024)
  • [CM1] R. Coifman and Y. Meyer, On commutators of singular integral and bilinear singular integrals, Trans. Amer. Math. Soc., 212 (1975), 315-331. MR 0380244 (52:1144)
  • [CM2] R. Coifman and Y. Meyer, Au del $ \grave{\rm a}$ des opérateurs pseudo-différentiels, Astérisque, 57 (1978). MR 518170 (81b:47061)
  • [CM3] R. Coifman and Y. Meyer, Ondelettes et opérateurs, III, Hermann, Paris, 1990. MR 1160989 (93i:42004)
  • [DM] X.T. Duong and A. McIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana, 15 (1999), 233-265. MR 1715407 (2001e:42017a)
  • [G] L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education, Inc., Upper Saddle River, New Jersey, 2004. MR 2449250
  • [GK] L. Grafakos and N. Kalton, Some remarks on multilinear maps and interpolation, Math. Ann., 319 (2001), 151-180. MR 1812822 (2002a:46032)
  • [GL] L. Grafakos and X. Li, Uniform bounds for the bilinear Hilbert transforms, I, Ann. of Math. (2), 159 (2004), 889-933. MR 2113017 (2006e:42011)
  • [GT1] L. Grafakos and R.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). Publ. Mat. (2002), Extra, 57-91. MR 1964816 (2004c:42031)
  • [GT2] L. Grafakos and R.H. Torres, Multilinear Calderón-Zygmund theory, Adv. in Math., 165 (2002), 124-164. MR 1880324 (2002j:42029)
  • [FS] C. Fefferman and E.M. Stein, Some maximal inequalities, Amer. J. Math., 93 (1971), 107-115. MR 0284802 (44:2026)
  • [KS] C. E. Kenig and E.M. Stein, Multilinear estimates and fractional integration, Math. Res. Letters, 6 (1999), 1-15. MR 1682725 (2000k:42023a)
  • [LT1] M. Lacey and C. Thiele, $ L^p$ estimates on the bilinear Hilbert transform for $ 2<p<\infty,$ Ann. of Math. (2), 146 (1997), 693-724. MR 1491450 (99b:42014)
  • [LT2] M. Lacey and C. Thiele, On Calderón's conjecture, Ann. of Math. (2), 149 (1999), 475-496. MR 1689336 (2000d:42003)
  • [L] X. Li, Uniform bounds for the bilinear Hilbert transforms, II, Rev. Mat. Iberoamericana, 22 (2006), 1069-1126. MR 2320411 (2008c:42014)
  • [M] T. Murai, A real variable method for the Cauchy transforms, and analytic capacity, Lecture Notes in Math., 1307, Springer, Berlin, 1988. MR 944308 (89k:30022)
  • [St] E.M. Stein, Harmonic Analysis: Real variable methods, orthogonality and oscillatory integrals, Princeton Univ. Press, 1993. MR 1232192 (95c:42002)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42B20, 42B25, 46B70, 47G30

Retrieve articles in all journals with MSC (2000): 42B20, 42B25, 46B70, 47G30


Additional Information

Xuan Thinh Duong
Affiliation: Department of Mathematics, Macquarie University, NSW, 2109, Australia
Email: duong@ics.mq.edu.au

Loukas Grafakos
Affiliation: Department of Mathematics, University of Missouri, Columbia, Missouri 65211
Email: loukas@math.missouri.edu

Lixin Yan
Affiliation: Department of Mathematics, Zhongshan University, Guangzhou, 510275, People’s Republic of China
Email: mcsylx@mail.sysu.edu.cn

DOI: https://doi.org/10.1090/S0002-9947-09-04867-3
Keywords: Multilinear operators, approximation to the identity, generalized Calder\'on-Zygmund kernel, Calder\'on-Zygmund decomposition, commutators
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).
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society