Solution to a conjecture on the norm of the Hardy operator minus the identity

We prove that for a decreasing weight w, the following inequality is sharp: ∞ 0 f ∗∗ (t)− f ∗ (t) p w(t) dt w Bp ∞ 0 f ∗ (t) p w(t) dt, where Bp is the Ariño and Muckenhoupt class of weights, and p 2. The case w ≡ 1 gives a positive answer to a conjecture formulated in Kruglyak and Setterqvist (2008) [8], where this estimate is proved only when p 2 is an integer. Simple examples show that, for 1 < p < 2, or if w is not decreasing, the result is false. Finally, using a different argument, we also prove that in the case p = 1, and for arbitrary weights w ∈ B1, w B1 is the best constant in the corresponding inequality. © 2010 Elsevier Inc. All rights reserved.