Averages over surfaces with infinitely flat points

J.G. Bak

Let $S$ be a hypersurface in $\bR^n$, $n\geq 2$, and let $d\mu=\psi d\sigma$, where $\psi\in C_0^{\infty} (\bR^n)$ and $\sigma$ denotes the surface area measure on $S$. Define the maximal function $\CM$ associated to $S$ and $\m$ by

$$\CM f(x) =\sup\limits_{t>0} \bigg|\int_S f(x-t\xi)d\mu(\xi)\bigg|, \qquad f\in\calS(\bR^n).$$

It was shown by E. M. Stein that when $S$ is the sphere in $\bR^n$, $n\geq 3$, $\CM$ (the spherical maximal function) is bounded on $L^p(\bR^n)$ if and only if $p>n/(n-1)$. It has also been shown that if $S$ is of finite type, i.e. the curvature vanishes to at most a finite order $m$ at every point of $S$, then there exists some number $p_m <\infty$ such that $\CM$ is bounded on $L^p(\bR^n)$ ($n\geq 3$) for all $p \in (p_m, \infty]$. On the other hand it is well known that if $S$ is {\it flat}, that is, $S$ contains a point at which the curvature vanishes to infinite order, then $\CM$ may not be bounded on any $L^p(\bR^n)$, $p<\infty$. We show that under some hypotheses the maximal functions $\CM$ associated to flat surfaces $S\subset \bR^3$ are bounded on certain Orlicz spaces $L^{\Phi}(\bR^3)$ defined naturally in terms of $S$.