A remark on the derivative of the one-dimensional Hardy-Littlewood maximal function

Abstract

J. Kinnunen proved that if p > 1, d ≤ 1 and f is a function in the Sobolev space W1,p (Rd), then the first order weak partial derivatives of the Hardy-Littlewood maximal function Mf belong to Lp(Rd). We shall show that, when d = 1, Kinnunen's result can be extended to the case where p = 1.