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) R, and the maximal function is localized in (0; 1). Moreover, we prove that the inequality kMfkp);w ckfkp);w holds with some c inde- pendent of f i w belongs to the well known Muckenhoupt class Ap, and therefore i kMfkp;w ckfkp;w for some c independent of f.