Hardy's operator minus identity and power weights

Abstract

Let H be the Hardy operator and I the identity operator acting on functions on the real half-line. We find optimal bounds for the operator H − I in the setting of power weights and the cases of positive decreasing functions, positive functions, and general functions. As a byproduct, we obtain some results about the optimal relations between the norms of H and its dual. ∞ 0 |f (t)| p dt.