The maximal theorem for weighted grand Lebesgue spaces

We study the Hardy's inequality and derive the maximal theorem of Hardy and Littlewood in the context of the grand Lebesgue spaces, considered when the underlying measure space is the interval $(0,1)\subset\erre$, and the maximal function is localized in $(0,1)$. Moreover, we characterize the weights in order that the inequality $$\| Mf\|_{p),w}\le c\| f\|_{p),w}$$ holds, and we prove that such inequality is true for some $c$ independent of $f$ iff $w$ belongs to the well known Muckenhoupt class $A_p$, and therefore iff $$\| Mf\|_{p,w}\le c\| f\|_{p,w}$$ is true for some $c$ independent of $f$. Some results of similar type are discussed for the case of small Lebesgue spaces.