Plancherel-Polya 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-Polya type inequalities on spaces of homogeneous type. These inequalities give new characterizations of the Besov spaces over dot (B)(p)(alpha,q) and the Triebel-Lizorkin spaces over 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 less than or equal to 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.