On a weighted inequality of Hardy type in spaces \(L^{p(\cdot)}\)
The boundedness of the Hardy type operator \(Hf(x) = \int_{\{t\in \mathbb R^ n :|t| \leq |x|\}}f(t)\,dt\) in weighted variable exponent Lebesgue spaces \(L^{p(.)}\) is studied. In particular, necessary and sufficient conditions are established on the weight functions \(v(x), w(x)\) and the exponents \(p(x), q(x)\) for which the inequality \[ \|v^{\frac1{q(x)}}Hf(x)\|_{q(.)} \leq C\| w^{\frac1{p(x)}}f(x)\|_{p(.)} \] is valid. An application of the result obtained to power weights is also given.
scientific article; zbMATH DE number 5553633
scientific article