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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 \(A_{p^+}\) \(A_{p^+}\)