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)



Atomic and molecular decompositions of anisotropic Triebel-Lizorkin spaces

Authors: Marcin Bownik and Kwok-Pun Ho
Journal: Trans. Amer. Math. Soc. 358 (2006), 1469-1510
MSC (2000): Primary 42B25, 42B35, 42C40; Secondary 46E35, 47B37, 47B38
Published electronically: March 25, 2005
MathSciNet review: 2186983
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Weighted anisotropic Triebel-Lizorkin spaces are introduced and studied with the use of discrete wavelet transforms. This study extends the isotropic methods of dyadic $\varphi$-transforms of Frazier and Jawerth (1985, 1989) to non-isotropic settings associated with general expansive matrix dilations and $A_\infty$ weights.

In close analogy with the isotropic theory, we show that weighted anisotropic Triebel-Lizorkin spaces are characterized by the magnitude of the $\varphi$-transforms in appropriate sequence spaces. We also introduce non-isotropic analogues of the class of almost diagonal operators and we obtain atomic and molecular decompositions of these spaces, thus extending isotropic results of Frazier and Jawerth.

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

  • 1. K.F. Andersen, R.T. John, Weighted inequalities for vector-valued maximal functions and singular integrals, Studia Math. 69 (1980), 19-31.MR 0604351 (82b:42015)
  • 2. M.Z. Berkolaiko, I.Ya. Novikov, Unconditional bases in spaces of functions of anisotropic smoothness, Proc. Steklov Inst. Math. 204 (1994), 27-41.MR 1320017 (96c:46037)
  • 3. M.Z. Berkolaiko, I.Ya. Novikov, Wavelet bases and linear operators in anisotropic Lizorkin-Triebel spaces, Dokl. Akad. Nauk, 340 (1995), 583-586.MR 1327833 (96a:42039)
  • 4. O.V. Besov, V.P. Il'in, S.M. Nikol'skii, Integral representations of functions and imbedding theorems. Vol. I and II, V. H. Winston & Sons, Washington, D.C., 1979.MR 0519341 (80f:46030a); MR 0521808 (80f:46030b)
  • 5. M. Bownik, A characterization of affine dual frames in $L^2(\mathbb{R}^n)$, Appl. Comput. Harmon. Anal. 8 (2000), 203-221. MR 1743536 (2001d:42019)
  • 6. M. Bownik, Anisotropic Hardy spaces and wavelets, Mem. Amer. Math. Soc. 164 (2003), no. 781, 122 pp. MR 1982689 (2004e:42023)
  • 7. M. Bownik, Atomic and Molecular Decompositions of Anisotropic Besov Spaces, Math. Z. (to appear).
  • 8. H.-Q. Bui, Weighted Besov and Triebel spaces: interpolation by the real method, Hiroshima Math. J. 12 (1982), 581-605. MR 0676560 (84f:46038)
  • 9. H.-Q. Bui, Characterizations of weighted Besov and Triebel-Lizorkin spaces via temperatures, J. Funct. Anal. 55 (1984), 39-62.MR 0733032 (86a:46034)
  • 10. H.-Q. Bui, Weighted Young's inequality and convolution theorems on weighted Besov spaces, Math. Nachr. 170 (1994), 25-37. MR 1302364 (95i:46038)
  • 11. H.-Q. Bui, M. Paluszynski, M.H. Taibleson, A maximal function characterization of weighted Besov-Lipschitz and Triebel-Lizorkin spaces, Studia Math. 119 (1996), 219-246. MR 1397492 (97c:46040)
  • 12. H.-Q. Bui, M. Paluszynski, M.H. Taibleson, Characterization of the Besov-Lipschitz and Triebel-Lizorkin spaces. The case $q<1$, J. Fourier Anal. Appl. 3 (1997), 837-846. MR 1600199 (99d:46045)
  • 13. A.P. Calderón, An atomic decomposition of distribution in parabolic $H^{p}$ spaces, Adv. in Math. 25 (1977), 216-225. MR 0448066 (56:6376)
  • 14. A.P. Calderón, A. Torchinsky, Parabolic maximal function associated with a distribution, Adv. in Math. 16 (1975), 1-64. MR 0417687 (54:5736)
  • 15. A.P. Calderón, A. Torchinsky, Parabolic maximal function associated with a distribution II, Adv. in Math. 24 (1977), 101-171.MR 0450888 (56:9180)
  • 16. C. Chui, W. Czaja, M. Maggioni, G. Weiss, Characterization of general tight wavelet frames with matrix dilations and tightness preserving oversampling, J. Fourier Anal. Appl. 8 (2002), no. 2, 173-200. MR 1891728 (2003a:42038)
  • 17. R.R. Coifman, A real variable characterization of $H^p$, Studia Math. 51 (1974), 269-274. MR 0358318 (50:10784)
  • 18. R.R. Coifman, G. Weiss, Analyse harmonique non-commutative sur certains espaces homogenes, Lecture Notes in Math., #242, Springer-Verlag (1971).MR 0499948 (58:17690)
  • 19. R.R. Coifman, G. Weiss, Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977), 569-645. MR 0447954 (56:6264)
  • 20. H. Dappa, W. Trebels, On anisotropic Besov and Bessel potential spaces, Approximation and function spaces (Warsaw, 1986), 69-87, PWN, Warsaw (1989). MR 1097182 (92b:46038)
  • 21. W. Farkas, Atomic and subatomic decompositions in anisotropic function spaces, Math. Nachr. 209 (2000), 83-113. MR 1734360 (2001h:46049)
  • 22. C. Fefferman, E.M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. MR 0284802 (44:2026)
  • 23. C. Fefferman, E.M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), 137-193. MR 0447953 (56:6263)
  • 24. G.B. Folland, E.M. Stein, Hardy spaces on homogeneous groups, Princeton University Press, Princeton, N.J., 1982. MR 0657581 (84h:43027)
  • 25. M. Frazier, B. Jawerth, Decomposition of Besov spaces, Indiana U. Math. J. 34 (1985), 777-799. MR 0808825 (87h:46083)
  • 26. M. Frazier, B. Jawerth, The $\varphi$-transform and applications to distribution spaces, Lecture Notes in Math., #1302, Springer-Verlag (1988), 223-246.MR 0942271 (89g:46064)
  • 27. M. Frazier, B. Jawerth, A Discrete Transform and Decomposition of Distribution Spaces, J. Funct. Anal. 93 (1989), 34-170. MR 1070037 (92a:46042)
  • 28. M. Frazier, B. Jawerth, G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conference Ser., #79, American Math. Society (1991).MR 1107300 (92m:42021)
  • 29. J. García-Cuerva, J.L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland (1985). MR 0807149 (87d:42023)
  • 30. I. Genebashvili, A. Gogatishvili, V. Kokilashvili, M. Krbec, Weight theory for integral transforms on spaces of homogeneous type, Longman, Harlow (1998).MR 1791462 (2003b:42002)
  • 31. J. Gilbert, Y. Han, J. Hogan, J. Lakey, D. Weiland, G. Weiss, Smooth molecular decompositions of functions and singular integral operators, Mem. Amer. Math. Soc. 156 (2002). MR 1880991 (2003f:42026)
  • 32. K.-P. Ho, Anisotropic Function spaces, Ph.D. Dissertation, Washington University (2002).
  • 33. P.-G. Lemarié-Rieusset, Projecteurs invariants, matrices de dilatation, ondelettes et analyses multi-résolutions Rev. Mat. Iberoamericana 10 (1994), 283-347. MR 1286477 (95e:42039)
  • 34. Y. Meyer, Wavelets and operators, Cambridge University Press, Cambridge (1992).MR 1228209 (94f:42001)
  • 35. Y. Meyer, R. Coifman, Wavelets. Calderón-Zygmund and multilinear operators, Cambridge University Press, Cambridge (1997). MR 1456993 (98e:42001)
  • 36. J. Peetre, New thoughts on Besov spaces, Duke University Mathematics Series, No. 1, Mathematics Department, Duke University, Durham, N.C., 1976.MR 0461123 (57:1108)
  • 37. A. Seeger, A note on Triebel-Lizorkin spaces, Approximation and function spaces (Warsaw, 1986), 391-400, PWN, Warsaw (1989). MR 1097208 (92c:46043)
  • 38. E.M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Univ. Press (1993). MR 1232192 (95c:42002)
  • 39. H.-J. Schmeisser, H. Triebel, Topics in Fourier Analysis and Function Spaces, John Wiley & Sons (1987). MR 0891189 (88k:42015b)
  • 40. J.-O. Strömberg, A. Torchinsky, Weighted Hardy Spaces, Lecture Notes in Math., #1381, Springer-Verlag (1989). MR 1011673 (90j:42053)
  • 41. W. Szlenk, An introduction to the theory of smooth dynamical systems, Translated from the Polish by Marcin E. Kuczma, PWN--Polish Scientific Publishers, Warsaw (1984). MR 0791919 (86f:58042)
  • 42. H. Triebel, Theory of Function Spaces, Monographs in Math., #78, Birkhäuser (1983).MR 0781540 (86j:46026)
  • 43. H. Triebel, Theory of function spaces II, Monographs in Math., #84, Birkhäuser Verlag, Basel (1992). MR 1163193 (93f:46029)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 42B25, 42B35, 42C40, 46E35, 47B37, 47B38

Retrieve articles in all journals with MSC (2000): 42B25, 42B35, 42C40, 46E35, 47B37, 47B38

Additional Information

Marcin Bownik
Affiliation: Department of Mathematics, University of Michigan, 525 East University Ave., Ann Arbor, Michigan 48109
Address at time of publication: Department of Mathematics, University of Oregon, Eugene, Oregon 97403–1222

Kwok-Pun Ho
Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong (China)

Keywords: Anisotropic Triebel-Lizorkin space, smooth atomic decomposition, smooth molecular decomposition, almost diagonal operators, wavelets, $\varphi$-transform, discrete wavelet transform
Received by editor(s): April 16, 2003
Received by editor(s) in revised form: March 8, 2004
Published electronically: March 25, 2005
Additional Notes: The first author was partially supported by NSF grant DMS-0200080
The authors thank Michael Frazier for careful reading and several suggestions for improvement of the paper, and Guido Weiss for making this joint work possible
Article copyright: © Copyright 2005 American Mathematical Society

American Mathematical Society