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Proceedings of the London Mathematical Society Advance Access published online on April 2, 2008
Proceedings of the London Mathematical Society, doi:10.1112/plms/pdn010
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© 2008 London Mathematical Society

Decomposition of weighted Triebel–Lizorkin and Besov spaces on the ball

G. Kyriazis

Department of Mathematics and Statistics
University of Cyprus
1678 Nicosia
Cyprus

P. Petrushev

Department of Mathematics
University of South Carolina
Columbia, SC 29205
USA
pencho@math.sc.edu

Yuan Xu

Department of Mathematics
University of Oregon
Eugene, Oregon 97403
USA
yuan@math.uoregon.edu

Received 15 February 2007. Revision received 22 January 2008.

Weighted Triebel–Lizorkin and Besov spaces on the unit ball Bd in Rd with weights wµ(x)=(1–|x|2)µ–1/2, µ≥0, are introduced and explored. A decomposition scheme is developed in terms of almost exponentially localized polynomial elements (needlets) {{varphi}{xi}}, {{psi}{xi}} and it is shown that the membership of a distribution to the weighted Triebel–Lizorkin or Besov spaces can be determined by the size of the needlet coefficients {<f, {varphi}{xi}>} in appropriate sequence spaces.

The second author has been supported by NSF grant DMS-0709046 and the third author by NSF grant DMS-0604056.

2000 Mathematics Subject Classification 41A25, 42B35, 42C15.


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