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Cruz-Uribe, David; Wang, Li-An Daniel

Extrapolation and weighted norm inequalities in the variable Lebesgue spaces. (English) Zbl 1354.42028

Trans. Am. Math. Soc. 369, No. 2, 1205-1235 (2017).
The variable Lebesgue space \(L^{p(\cdot)}\) has been the subject of considerable interest since the early 1990s. In this paper a generalization of the Rubio de Francia extrapolation theorem to weighted variable Lebesgue spaces \(L^{p(\cdot)}\) is investigated. D. Cruz-Uribe et al. [J. Funct. Anal. 213, No. 2, 412–439 (2004; Zbl 1052.42016)] proved the following. If for some \(p_0 >0\) and every \(w_0 \in A_{\infty}\), \[ \int_{{\mathbb R}^n} f(x_0)^{p_0} w_0(x) \, dx \leq C \int_{{\mathbb R}^n} g(x_0)^{p_0} w_0(x) \, dx, \] then the same inequality holds with \(p_0\) replaced by any \(p, 0<p<\infty\).
The authors consider this theorem on the variable Lebesgue spaces. They further develop the theory of weighted norm inequalities on \(L^{p(\cdot)}\), and prove weighted boundedness of several operators (ex. singular integrals, fractional integrals and vector valued operators).
Reviewer: Yasuo Komori-Furuya (Kanagawa)

Cited in 1 Review
Cited in 60 Documents

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B35 Function spaces arising in harmonic analysis

Keywords:

variable Lebesgue spaces; Muckenhoupt weight; maximal operator; singular integral; fractional integral; Rubio de Francia extrapol

Citations:

Zbl 1052.42016
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Full Text: DOI arXiv

References:

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