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)

 
 

 

Discrete decompositions for bilinear operators and almost diagonal conditions


Authors: Loukas Grafakos and Rodolfo H. Torres
Journal: Trans. Amer. Math. Soc. 354 (2002), 1153-1176
MSC (1991): Primary 42B25, 42B20, 47G30; Secondary 42C15, 46E35, 35S99
DOI: https://doi.org/10.1090/S0002-9947-01-02912-9
Published electronically: October 24, 2001
MathSciNet review: 1867376
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Using discrete decomposition techniques, bilinear operators are naturally associated with trilinear tensors. An intrinsic size condition on the entries of such tensors is introduced and is used to prove boundedness for the corresponding bilinear operators on several products of function spaces. This condition should be considered as the direct analogue of an almost diagonal condition for linear operators of Calderón-Zygmund type. Applications include a reduced $T1$ theorem for bilinear pseudodifferential operators and the extension of an $L^p$ multiplier result of Coifman and Meyer to the full range of $H^p$ spaces. The results of this article rely on decomposition techniques developed by Frazier and Jawerth and on the vector valued maximal function estimate of Fefferman and Stein.


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

  • 1. G. Bourdaud, Une algèbre maximale d' opérateurs pseudodifferentiels, Comm. Partial Diff. Eq. 13 (1988), 1059-1083. MR 89g:47063
  • 2. R. R. Coifman, S. Dobyinsky, and Y. Meyer, Opérateurs bilinéaires et renormalization, in Essays on Fourier Analysis in Honor of Elias M. Stein, C. Fefferman, R. Fefferman, S. Wainger (eds), Princeton University Press, Princeton NJ, 1995. MR 96b:42019
  • 3. R. R. Coifman and L. Grafakos, Hardy space estimates for multilinear operators, Rev. Mat. Iberoamer. 8 (1992), 45-67. MR 93j:42011
  • 4. R. R. Coifman, P.-L. Lions, Y. Meyer, and S. Semmes, Compensated compactness and Hardy spaces, J. Math. Pures et Appl. 72 (1993), 247-286. MR 95d:46033
  • 5. R. R. Coifman and Y. Meyer, Au delà des opérateurs pseudo-différentiels, Astérisque 57, 1978. MR 81b:47061
  • 6. R. R. Coifman and Y. Meyer Wavelets: Calderón-Zygmund and multilinear operators Cambridge Univ. Press, Cambridge, United Kingdom, 1997. MR 98e:42001
  • 7. G. David and J-L. Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. 120 (1984), 371-397. MR 85k:42041
  • 8. G. David, J-L. Journé, and S. Semmes, Opérateurs de Calderón-Zygmund, fonctions para-accrétives et interpolation, Rev. Mat. Iber. 1 (1985), 1-56. MR 88f:47024
  • 9. C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. MR 44:2026
  • 10. M. Frazier, B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Func. Anal. 93 (1990), 34-169. MR 92a:46042
  • 11. M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley theory and the study of function spaces, CBMS Regional Conference Series in Mathematics 79, 1991. MR 92m:42021
  • 12. J. Gilbert and A. Nahmod, Hardy spaces and a Walsh model for bilinear cone operators, Trans. Amer. Math. Soc. 351 (1999), 3267-3300. MR 99m:42022
  • 13. L. Grafakos and R. Torres, Pseudodifferential operators with homogeneous symbols, Michigan Math. J. 46 (1999), 261-269. MR 2000h:35169
  • 14. L. Grafakos and R. Torres, A multilinear Schur test and multiplier operators, to appear in J. Func. Anal.
  • 15. L. Hörmander, Pseudodifferential operators of type 1,1, Comm. Partial Diff. Eq. 13 (1988), 1085-1111. MR 89k:35260
  • 16. L. Hörmander, Continuity of pseudodifferential operators of type 1,1, Comm. Partial Diff. Eq. 14 (1989), 231-243. MR 90a:35241
  • 17. M. Lacey and C. Thiele, $L^p$ estimates on the bilinear Hilbert transform for $2<p<\infty$, Ann. of Math. 146 (1997), 693-724. MR 99b:42014
  • 18. Y. Meyer, Regularité des solutions des équations aux dérivées partielles non linéaires, Springer-Verlag Lectures Notes in Math. 842 (1981), 293-302. MR 84c:35015
  • 19. Y. Meyer, Ondelettes et opérateurs I and II, Hermann, Paris, 1990. MR 93i:42002; MR 93i:42003
  • 20. E. M. Stein, Harmonic analysis: Real variable methods, orthogonality, and oscillatory integrals, Princeton University Press, Princeton NJ, 1993. MR 95c:42002
  • 21. R. Torres, Boundedness results for operators with singular kernels on distribution spaces, Mem. Amer. Math. Soc. 442, 1991. MR 91g:47044
  • 22. H. Triebel, Theory of function spaces, Monographs in Mathematics, Vol. 78, Birkhauser Verlag, Basel, 1983. MR 86j:46026
  • 23. A. Youssfi, Bilinear operators and the Jacobian-determinant on Besov spaces, Indiana Univ. Math. J. 45 (1996), 381-396. MR 97h:46050

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 42B25, 42B20, 47G30, 42C15, 46E35, 35S99

Retrieve articles in all journals with MSC (1991): 42B25, 42B20, 47G30, 42C15, 46E35, 35S99


Additional Information

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

Rodolfo H. Torres
Affiliation: Department of Mathematics, University of Kansas, Lawrence, Kansas 66049
Email: torres@math.ukans.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02912-9
Keywords: Singular integrals, maximal functions, Littlewood-Paley theory, almost diagonal condition, multilinear operators, wavelets, Triebel-Lizorkin spaces
Received by editor(s): May 20, 1999
Published electronically: October 24, 2001
Additional Notes: Grafakos’ research partially supported by the NSF under grant DMS 9623120
Torres’ research partially supported by the NSF under grant DMS 9696267
Article copyright: © Copyright 2001 American Mathematical Society

American Mathematical Society