A Necessary and Sufficient Condition for Hardy’s Operator in the Variable Lebesgue Space

Abstract

The variable exponent Hardy inequality ${∥x^{\beta (x)-1}{\int }_0^x\mathrm{‍}f(t)dt∥}_{L^{p(.)}(0,l)}\le C{∥x^{\beta (x)}f∥}_{L^{p(.)}(0,l)}$, $f\ge 0$ is proved assuming that the exponents $p:(0,l)\to (1,\infty )$, $\beta :(0,l)\to ℝ$ not rapidly oscilate near origin and $1/p^{\prime }(0)-\beta >0$. The main result is a necessary and sufficient condition on $p$, $\beta$ generalizing known results on this inequality.