Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Plancherel-Pôlya type inequality on spaces
of homogeneous type and its applications

Author: Y.-S. Han
Journal: Proc. Amer. Math. Soc. 126 (1998), 3315-3327
MSC (1991): Primary 42B25, 46F05
MathSciNet review: 1459123
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Abstract: In this paper, using the discrete Calderon reproducing formula on spaces of homogeneous type obtained by the author, we obtain the Plancherel-Pôlya type inequalities on spaces of homogeneous type. These inequalities give new characterizations of the Besov spaces $\dot B_p^{\alpha ,q}$ and the Triebel-Lizorkin spaces $\dot F_p^{\alpha ,q}$ on spaces of homogeneous type introduced earlier by the author and E. T. Sawyer and also allow us to generalize these spaces to the case where $p,q\le 1$. Moreover, using these inequalities, we can easily show that the Littlewood-Paley $G$-function and $S$-function are equivalent on spaces of homogeneous type, which gives a new characterization of the Hardy spaces on spaces of homogeneous type introduced by Macias and Segovia.

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Y.-S. Han
Affiliation: Department of Mathematics, Auburn University, Auburn, Alabama 36849-5310

Keywords: Plancherel-P\^olya type inequality, spaces of homogeneous type, Besov and Triebel-Lizorkin spaces, Littlewood-Paley $G$-function and $S$-function, discrete Calderon formula
Received by editor(s): September 19, 1996
Received by editor(s) in revised form: April 1, 1997
Communicated by: J. Marshall Ash
Article copyright: © Copyright 1998 American Mathematical Society