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

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.