The maximal theorem for weighted grand Lebesgue spaces

We study the Hardy inequality and derive the maximal theorem of Hardy and Littlewood in the context of grand Lebesgue spaces, considered when the underlying measure space is the interval (0, 1) subset of R, and the maximal function is localized in (0, 1). Moreover, we prove that the inequality parallel to Mf parallel to(p),w) <= c parallel to f parallel to(p),w) holds with some c independent of f iff w belongs to the well known Muckenhoupt class A(p), and therefore iff parallel to Mf parallel to(p,w) <= c parallel to f parallel to(p,w) for some c independent of f. Some results of similar type are discussed for the case of small Lebesgue spaces.