Title: Estimates for maximal functions associated to hypersurfaces in $\Bbb R^3$ with height $h<2:$ Part I,
Abstract: In this article, we continue the study of the problem of $L^p$-boundedness of
the maximal operator $M$ associated to averages along isotropic dilates of a
given, smooth hypersurface $S$ of finite type in 3-dimensional Euclidean space.
An essentially complete answer to this problem had been given about seven years
ago by the last named two authors in joint work with M. Kempe for the case
where the height h of the given surface is at least two. In the present
article, we turn to the case $h<2.$ More precisely, in this Part I, we study
the case where $h<2,$ assuming that $S$ is contained in a sufficiently small
neighborhood of a given point $x^0\in S$ at which both principal curvatures of
$S$ vanish. Under these assumptions and a natural transversality assumption, we
show that, as in the case where $h\ge 2,$ the critical Lebesgue exponent for
the boundedness of $M$ remains to be $p_c=h,$ even though the proof of this
result turns out to require new methods, some of which are inspired by the more
recent work by the last named two authors on Fourier restriction to S. Results
on the case where $h<2$ and exactly one principal curvature of $S$ does not
vanish at $x^0$ will appear elsewhere.