A necessary and sufficient condition for Hardy's operator in the variable Lebesgue space
Summary: The variable exponent Hardy inequality \(\|x^{\beta(x) - 1} \int_0^x f(t) d t\|_{L^{p(.)}(0, l)} \leq C \|x^{\beta(x)} f\|_{L^{p(.)}(0, l)}\), \(f \geq 0\) is proved assuming that the exponents \(p :(0, l) \rightarrow(1, \infty)\), \(\beta :(0, l) \rightarrow \mathbb{R}\) not rapidly oscilate near origin and \(1 / p'(0) - \beta > 0\). The main result is a necessary and sufficient condition on \(p\), \(\beta\) generalizing known results on this inequality.
scientific article; zbMATH DE number 7022194
scientific article