The maximal theorem for weighted grand Lebesgue spaces

Abstract

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 ||M/f||p),w, ≤ c||f||p),w, holds with some c independent of f iff w belongs to the well known Muckenhoupt class Ap, and therefore iff ||Mf||p,w ≤ c||f||p,w for some c independent of f. Some results of similar type are discussed for the case of small Lebesgue spaces. © Instytut Matematyczny PAN, 2008.