Weighted fully measurable grand Lebesgue spaces and the maximal theorem

Abstract

Anatriello and Fiorenza (J Math Anal Appl 422:783–797, 2015) introduced the fully measurable grand Lebesgue spaces on the interval $(0,1)\subset \mathbb R$, which contain some known Banach spaces of functions, among which there are the classical and the grand Lebesgue spaces, and the $EXP_\alpha$ spaces $(\alpha >0)$. In this paper we introduce the weighted fully measurable grand Lebesgue spaces and we prove the boundedness of the Hardy–Littlewood maximal function. Namely, let

$$\begin{aligned} \Vert f\Vert _ {p[\cdot ],\delta (\cdot ), w}={{\mathrm{ess\,sup}}}_{x\in (0,1)} \left( \int _0^1 (\delta (x)f(t))^{p(x)} w(t)\mathrm{dt}\right) ^{\frac{1}{p(x)}}, \end{aligned}$$

where w is a weight, $0<\delta (\cdot )\le 1\le p(\cdot )<\infty$, we show that if $\displaystyle {p^+}=\Vert p\Vert _\infty <+\infty$, the inequality

$$\begin{aligned}\Vert Mf\Vert _{p[\cdot ],\delta (\cdot ),w} \le c\Vert f\Vert _{p[\cdot ],\delta (\cdot ),w} \end{aligned}$$

holds with some constant c independent of f if and only if the weight w belongs to the Muckenhoupt class $A_{p^+}$.