Weight characterization of an averaging operator

Let 0<α<1 and , x?0. A factorization theorem is given, which provides a weight characterization of the space of all positive functions f such that T_αf belongs to L^p_w, 1<p<∞, w a weight function. This theorem yields a two-sided estimate for the norm of T_αf. An analogous result holds for α=0. In the latter case, it is also shown that the averaging Hardy operator T₀ and its dual  are comparable in L^p_w, 1<p<∞, if w belongs to the Muckenhoupt weight class A_p.