The Hardy-Littlewood maximal type operators between Banach function spaces (Q2841844)

The main motivation of the paper is to study the two weight problem for the Hardy-Littlewood maximal operator within the context of Banach function spaces. The authors work with a general variant of the maximal operator \(M_X\) generated by a rearrangement invariant (r.i.) space on \(\mathbb{R}^N\). The boundedness of this type of operator between Banach function spaces is considered.NEWLINENEWLINEIn particular, by means of estimates of the rearrangement of the Hardy-Littlewood maximal function generated by a Lorentz space, it is provided a sufficient condition on the r.i. spaces \(F\) and \(X\) defined on \(\mathbb{R}^N\) to ensure that the maximal operator \(M_F\) is bounded on \(X\). As a corollary, the authors prove that if \(F\) is a r.i. space whose fundamental function \(\varphi_F\) is submultiplicative near zero, the maximal operator \(M_F\) is bounded on \(L^p(\mathbb{R}^N)\) if and only if NEWLINE\[NEWLINE \int_0^1 s^{-1/p} \;d\varphi_F(s) < +\infty. NEWLINE\]NEWLINE As an application, the authors give new sufficient conditions for the boundedness of the Hardy-Littlewood maximal operator between weighted \(L^p\)-spaces with different weights.