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Transactions of the American Mathematical Society
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
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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.


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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
Email: mbownik@uoregon.edu

Kwok-Pun Ho
Affiliation: Department of Mathematics, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong (China)
Email: makho@ust.hk

DOI: http://dx.doi.org/10.1090/S0002-9947-05-03660-3
PII: S 0002-9947(05)03660-3
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