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 inspire 查看全文>>