The Kakeya maximal operator on the variable Lebesgue spaces

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.