In this article, after establishing weighted Plancherel-Polya-type inequalities, we introduce a new class of weighted Hardy spaces H(b,w)(p) by using g-function, where w is a Muckenhoupt's weight and b is a para-accretive function. Then we show the atomic decomposition and molecular characterization of H(b,w)(p). As applications, we prove the boundedness of Calderon-Zygmund operators between H(b,w)(p) and classical weighted Hardy spaces H(w)(p).
