𝑇1 theorems for Besov and Triebel-Lizorkin spaces

Han, Y.-S.; Hofmann, Steve

doi:10.1090/S0002-9947-1993-1097168-4

$\textrm {T}1$ theorems for Besov and Triebel-Lizorkin spaces
HTML articles powered by AMS MathViewer

by Y.-S. Han and Steve Hofmann

Trans. Amer. Math. Soc. 337 (1993), 839-853

DOI: https://doi.org/10.1090/S0002-9947-1993-1097168-4

PDF | Request permission

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

Kenneth F. Andersen and Russel T. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math. 69 (1980/81), no. 1, 19–31. MR 604351, DOI 10.4064/sm-69-1-19-31
R. R. Coifman and C. Fefferman, Weighted norm inequalities for maximal functions and singular integrals, Studia Math. 51 (1974), 241–250. MR 358205, DOI 10.4064/sm-51-3-241-250
R. Coifman, G. David, Y. Meyer, and S. Semmes, $\omega$-Calderón-Zygmund operators, Harmonic analysis and partial differential equations (El Escorial, 1987) Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 132–145. MR 1013820, DOI 10.1007/BFb0086798
Guy David and Jean-Lin Journé, A boundedness criterion for generalized Calderón-Zygmund operators, Ann. of Math. (2) 120 (1984), no. 2, 371–397. MR 763911, DOI 10.2307/2006946
Michael Frazier and Björn Jawerth, A discrete transform and decompositions of distribution spaces, J. Funct. Anal. 93 (1990), no. 1, 34–170. MR 1070037, DOI 10.1016/0022-1236(90)90137-A
M. Frazier, Y.-S. Han, B. Jawerth, and G. Weiss, The $T1$ theorem for Triebel-Lizorkin spaces, Harmonic analysis and partial differential equations (El Escorial, 1987) Lecture Notes in Math., vol. 1384, Springer, Berlin, 1989, pp. 168–181. MR 1013823, DOI 10.1007/BFb0086801
José García-Cuerva and José L. Rubio de Francia, Weighted norm inequalities and related topics, North-Holland Mathematics Studies, vol. 116, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 104. MR 807149
Y.-S. Han and E. T. Sawyer, Para-accretive functions, the weak boundedness property and the $Tb$ theorem, Rev. Mat. Iberoamericana 6 (1990), no. 1-2, 17–41. MR 1086149, DOI 10.4171/RMI/93

A weak molecule condition and

Theorem for certain Triebel-Lizorkin spaces

Steve Hofmann, A weak molecule condition for certain Triebel-Lizorkin spaces, Studia Math. 101 (1992), no. 2, 113–122. MR 1149566, DOI 10.4064/sm-101-2-113-122

Calderón-Zygmund operators, pseudo-differential operators, and the Cauchy integral of Calderón

Douglas S. Kurtz and Richard L. Wheeden, Results on weighted norm inequalities for multipliers, Trans. Amer. Math. Soc. 255 (1979), 343–362. MR 542885, DOI 10.1090/S0002-9947-1979-0542885-8
Pierre Gilles Lemarié, Continuité sur les espaces de Besov des opérateurs définis par des intégrales singulières, Ann. Inst. Fourier (Grenoble) 35 (1985), no. 4, 175–187 (French, with English summary). MR 812324, DOI 10.5802/aif.1033

Continuité Besov de certains opérateurs intégraux singuliers

La minimalité de le espace de Besov

et la continuité de opérateurs définis par des intégrales singulières

Kôzô Yabuta, Generalizations of Calderón-Zygmund operators, Studia Math. 82 (1985), no. 1, 17–31. MR 809770, DOI 10.4064/sm-82-1-17-31
Yong Sheng Han, Björn Jawerth, Mitchell Taibleson, and Guido Weiss, Littlewood-Paley theory and $\epsilon$-families of operators, Colloq. Math. 60/61 (1990), no. 1, 321–359. MR 1096383, DOI 10.4064/cm-60-61-1-321-359