PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 128, Number 1, Pages 65–74 S 0002-9939(99)05128-X Article electronically published on June 30, 1999 REARRANGEMENT OF HARDY-LITTLEWOOD MAXIMAL FUNCTIONS IN LORENTZ SPACES JES ´ US BASTERO, MARIO MILMAN, AND FRANCISCO J. RUIZ (Communicated by Frederick W. Gehring) Abstract. For the classical Hardy-Littlewood maximal function Mf , a well known and important estimate due to Herz and Stein gives the equivalence (Mf ) ∗ (t) ∼ f ∗∗ (t). In the present note, we study the validity of analogous estimates for maximal operators of the form Mp,q f (x) = sup x∈Q ‖fχ Q ‖p,q ‖χ Q ‖p,q , where ‖.‖p,q denotes the Lorentz space L(p, q)-norm. 1. Introduction The Hardy-Littlewood maximal function M plays a central role in classical har- monic analysis, differentiation theory and PDE’s. It is well known that the maximal operator M is of weak type (1, 1) and strong type (∞, ∞) from where it follows readily, for example using K-functionals (see [BS]), that there exists an absolute constant C> 0 such that (Mf ) * (t) ≤ Cf ** (t), ∀t> 0, ∀f ∈ L 1 loc (R n ), (1) where * denotes non-increasing rearrangement, and f ** (t)= 1 t  t 0 f * (s)ds. Herz (cf. [H], and also [BS]) proved that the reverse inequality is also true, that is, f ** (t) ≤ c(Mf ) * (t),t> 0. (2) Inequalities (1) and (2) contain the basic information to study M , and the oper- ators it controls, in rearrangement invariant function spaces. We refer to [AKMP] for a recent and exhaustive study of inequalities (1) and (2) where the underlying measure is more general than Lebesgue measure. Recall that the maximal operator M is defined by Mf (x) = sup x∈Q 1 |Q|  Q |f (t)|dt = sup x∈Q ‖fχ Q ‖ L 1 ‖χ Q ‖ L 1 . Received by the editors March 2, 1998. 1991 Mathematics Subject Classification. Primary 42B25, 46E30. Key words and phrases. Maximal functions, rearrangement inequalities, Lorentz spaces. The first author was partially supported by DGICYT PB94-1185. The third author was partially supported by DGICYT and IER. c 1999 American Mathematical Society 65 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use