  We shall verify the Kakeya (Nikodym) maximal operator $K_{N}$, $N\gg 1$, is
bounded on the variable Lebesgue space $L^{p(\cdot)}(\mathbb{R}^2)$ when the
exponent function $p(\cdot)$ is $N$-modified locally log-H\"{o}lder continuous
and log-H\"{o}lder continuous at infinity.
