# Weighted fully measurable grand Lebesgue spaces and the maximal theorem

Research paper by Giuseppina Anatriello, Maria Rosaria Formica

Indexed on: 12 Aug '16Published on: 01 Jun '16Published in: Ricerche di Matematica

#### Abstract

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 $$A_{p^+}$$ .AbstractAnatriello 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 $$A_{p^+}$$ .2015 $$(0,1)\subset \mathbb R$$ $$(0,1)\subset \mathbb R$$ $$EXP_\alpha$$ $$EXP_\alpha$$ $$(\alpha >0)$$ $$(\alpha >0)$$ \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} \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}w $$0<\delta (\cdot )\le 1\le p(\cdot )<\infty$$ $$0<\delta (\cdot )\le 1\le p(\cdot )<\infty$$ $$\displaystyle {p^+}=\Vert p\Vert _\infty <+\infty$$ $$\displaystyle {p^+}=\Vert p\Vert _\infty <+\infty$$ \begin{aligned}\Vert Mf\Vert _{p[\cdot ],\delta (\cdot ),w} \le c\Vert f\Vert _{p[\cdot ],\delta (\cdot ),w} \end{aligned} \begin{aligned}\Vert Mf\Vert _{p[\cdot ],\delta (\cdot ),w} \le c\Vert f\Vert _{p[\cdot ],\delta (\cdot ),w} \end{aligned}cfw $$A_{p^+}$$ $$A_{p^+}$$