Estimates for maximal functions associated to hypersurfaces in $\mathbb {R}^3$ 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 $\mathcal {M}$ 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 was given about eight years ago by the third and fourth authors in joint work with M. Kempe [Acta Math 204 (2010), pp. 151–271] 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 $h\ge 2,$ the critical Lebesgue exponent for the boundedness of $\mathcal {M}$ 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 third and fourth 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.