Maximal Averages over Certain Non-smooth and Non-convex Hypersurfaces

Abstract

We consider the maximal operators whose averages are taken over some non-smooth and non-convex hypersurfaces. For each $1 \leq i \leq d-1$, let $\phi_i \colon [-1,1] \to \mathbb{R}$ be a continuous function satisfying some derivative conditions, and let $\phi(y) = \sum_{i=1}^{d-1} \phi_i(y_i)$. We prove the $L^p$ boundedness of the maximal operators associated with the graph of $\phi$ which is a non-smooth and non-convex hypersurface in $\mathbb{R}^d$, $d \geq 3$.