Weight characterization of an averaging operator

Let 0<α<1 and Tα : f →(1/[(1−α)x])( x αx f ), x 0. A factorization theorem is given, which provides a weight characterization of the space of all positive functions f such that Tαf belongs to L p w, 1 < p<∞, w a weight function. This theorem yields a two-sided estimate for the norm of Tαf . An analogous result holds for α = 0. In the latter case, it is also shown that the averaging Hardy operator T0 and its dual T ∗ 0 are comparable in L p w, 1< p <∞, if w belongs to the Muckenhoupt weight class Ap.  2003 Elsevier Inc. All rights reserved.