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

Han, Y.-S.

doi:10.1090/S0002-9939-98-04445-1

Plancherel-Pôlya type inequality on spaces of homogeneous type and its applications
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by Y.-S. Han

Proc. Amer. Math. Soc. 126 (1998), 3315-3327

DOI: https://doi.org/10.1090/S0002-9939-98-04445-1

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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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