Extrapolation and weighted norm inequalities in the variable Lebesgue spaces (Q2833017)

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. \textit{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 \(w_0 \in A_{\infty}\), NEWLINE\[NEWLINE \int_{{\mathbb R}^n} f(x_0)^{p_0} w_0(x) \, dx \leq C \int_{{\mathbb R}^n} g(x_0)^{p_0} w_0(x) \, dx, NEWLINE\]NEWLINE then the same inequality holds with \(p_0\) replaced by any \(p, 0<p<\infty\).NEWLINENEWLINEThe 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).