It is well-known results proved by E.M. Stein (1976 for the case $n\ge3$) and J. Bourgain (1985 for the case $n=2$) that the spherical maximal operator is bounded on $L^p({\Bbb R}^n)$ if and only if $p>n/(n-1)$. Further, the bounded-ness problem had been considered by many authors for more general hypersurfaces. The problem is connected to the geometric properties of the surface.
There is still the open conjecture of E.M. Stein about connection between the decay rate of the associated oscillatory integrals and the exponent $p$ such that the maximal operator is bounded on $L^p({\Bbb R}^n)$. Following A.N. Varchenko we introduce notion of "height" of hyper-surface at a point. This notion allows us to obtain the sharp exponent of boundedness for the maximal operator defined by any smooth hypersurface in $L^p({\Bbb R}3)$.
In particular, we get a confirmation of the E.M. Stein conjecture for arbitrary two-dimensional smooth hypersurfaces. In the talk we also give some estimates for related oscillatory integrals with smooth phase functions.