On a dual property of the maximal operator on weighted variable $L^p$ spaces

L. Diening \cite{D1} obtained the following dual property of the maximal operator $M$ on variable Lebesque spaces $L^{p(\cdot )}$: if $M$ is bounded on $L^{p(\cdot )}$, then $M$ is bounded on $L^{p^{\mathrm{\prime }}(\cdot )}$. We extend this result to weighted variable Lebesque spaces.