On directional maximal operators associated with generalized lacunary sets

Abstract

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

References in corpus (1)

• An Estimate of the Maximal Operators Associated with Generalized Lacunary Sets