Rearrangement of Hardy-Littlewood maximal functions in Lorentz spaces

Abstract:

For the classical Hardy-Littlewood maximal function $Mf$, a well known and important estimate due to Herz and Stein gives the equivalence $(Mf)^{*}(t)\sim f^{**}(t)$. In the present note, we study the validity of analogous estimates for maximal operators of the form \begin{equation*}M_{p,q}f(x)= \sup _{x\in Q}{\frac {\Vert f\chi _{Q} \Vert _{p,q} }{\Vert \chi _{Q} \Vert _{p,q}}}, \end{equation*} where $\Vert . \Vert _{p,q}$ denotes the Lorentz space $L(p,q)$-norm.