On directional maximal operators associated with generalized lacunary sets

Let $\Omega$ be any set of directions (unit vectors) on the plane. We study maximal operators defined by \md0 M_\Omega f(x)=\sup_{\delta >0, \omega \in \Omega} \frac{1}{2\delta}\int_{-\delta}^\delta |f(x+t\omega)|dt. \emd for the generalized lacunary sets $\Omega$ associated with an integer $\mu >0$. It is proved the following sharp inequality: $$\|M_\Omega f(x)\|_2\lesssim \sqrt{\mu} \|f\|_2.$$