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)

 
 

 

$ {\rm T}1$ theorems for Besov and Triebel-Lizorkin spaces


Authors: Y.-S. Han and Steve Hofmann
Journal: Trans. Amer. Math. Soc. 337 (1993), 839-853
MSC: Primary 46E35; Secondary 42B20, 47G10
DOI: https://doi.org/10.1090/S0002-9947-1993-1097168-4
MathSciNet review: 1097168
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We give simple proofs of the $ T1$ theorem in the general context of Besov spaces and (weighted) Triebel-Lizorkin spaces. Our approach yields some new results for kernels satisfying weakened regularity conditions, while also recovering previously known results.


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

  • [AJ] K. F. Andersen and R. T. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math. 69 (1980), 19-31. MR 604351 (82b:42015)
  • [CF] R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241-250. MR 0358205 (50:10670)
  • [CDMS] R. Coifman, G. David, Y. Meyer, and S. Semmes, $ \omega $-Calderón-Zygmund operators, Proceedings of the Conference on Harmonic Analysis and P.D.E., El Escorial 1987, Lecture Notes in Math., vol. 1384, Springer-Verlag, Berlin, 1989, pp. 132-145. MR 1013820 (90k:42028)
  • [DJ] G. David and J. L. Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. (2) 120 (1984), 371-397. MR 763911 (85k:42041)
  • [FJ] M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J. Functional Anal. 93 (1990), 34-170. MR 1070037 (92a:46042)
  • [FJHW] M. Frazier, B. Jawerth, Y.-S. Han, and G. Weiss, The $ T1$ Theorem for Triebel-Lizorkin spaces, Proceedings of the Conference on Harmonic Analysis and P.D.E., El Escorial 1987, Lecture Notes in Math., vol. 1384, Springer-Verlag, Berlin, 1989. MR 1013823 (90j:46035)
  • [GR] J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Math. Studies, vol. 116, North-Holland, Amsterdam, 1985. MR 807149 (87d:42023)
  • [HS] Y. S. Han and E. T. Sawyer, Para-accretive functions, the weak boundedness property and the $ Tb$ Theorem, Rev. Mat. Iberoamericana 6 (1990), 17-41. MR 1086149 (92b:46041)
  • [HH] Y.-S. Han and S. Hofmann, A weak molecule condition and $ T1$ Theorem for certain Triebel-Lizorkin spaces, unpublished manuscript.
  • [H] S. Hofmann, A weak molecule condition for certain Triebel-Lizorkin spaces, Studia Math. 101 (1992), 113-122. MR 1149566 (93a:46057)
  • [J] J. L. Journé, Calderón-Zygmund operators, pseudo-differential operators, and the Cauchy integral of Calderón, Lecture Notes in Math., vol. 994, Springer-Verlag, Berlin, 1983.
  • [KW] D. S. Kurtz and R. L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979), 343-362. MR 542885 (81j:42021)
  • [L] P. G. Lemariè, Continuité sur les espaces de Besov des opérateurs définis par des intégrales singuliers, Ann. Inst. Fourier (Grenoble) 35 (1985), 175-187. MR 812324 (87j:47074)
  • [MM] M. Meyer, Continuité Besov de certains opérateurs intégraux singuliers, Thèse de 3e Cycle, Orsay 1985.
  • [M] Y. Meyer, La minimalité de le espace de Besov $ \dot B_1^{0,1}$, et la continuité de opérateurs définis par des intégrales singulières, Monografias de Matematicas, vol. 4, Univ. Autonoma de Madrid.
  • [Y] K. Yabuta, Generalizations of Calderón-Zygmund operators, Studia Math. 82 (1985), 17-31. MR 809770 (87e:47072)
  • [HJTW] Y. Han, B. Jawerth, M. Taibleson, and G. Weiss, Littlewood-Paley theory and $ \varepsilon $-families of operators, Colloq. Math. 60/61 (1990), 321-359 (added in proof). MR 1096383 (92h:46043)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46E35, 42B20, 47G10

Retrieve articles in all journals with MSC: 46E35, 42B20, 47G10


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1993-1097168-4
Article copyright: © Copyright 1993 American Mathematical Society

American Mathematical Society