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 \(A_{p^+}\).