Compactness for the weighted Hardy operator in variable exponent spaces

In this paper, we prove a necessary and sufficiency condition for the weighted Hardy operator

$H_{\upsilon ,\omega }f(x)=\upsilon (x)\underset{0}{\overset{x}{\int }}f(t)\omega (t)\mathrm{d}t$

to be compactly acting from $L^{p(?)}(0,\infty )$ to $L^{q(?)}(0,\infty )$.