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))pw(t)dt⩽‖w‖Bp∫0∞(f⁎(t))pw(t)dt, where BpBp is the Ariño and Muckenhoupt class of weights, and p⩾2p⩾2. The case w≡1w≡1 gives a positive answer to a conjecture formulated in Kruglyak and Setterqvist (2008) [8], where this estimate is proved only when p⩾2p⩾2 is an integer. Simple examples show that, for 1