Smooth and singular maximal averages over 2D hypersurfaces and associated Radon transforms

Abstract

We prove $L^p$ boundedness results, $p > 2$, for local maximal averaging operators over a smooth 2D hypersurface $S$ with either a $C^1$ density function or a density function with a singularity that grows as $|(x,y)|^{-\beta}$ for $\beta < 2$. Suppose one is in coordinates such that the surface is localized near some $(x_0,y_0,z_0)$ at which $(0,0,1)$ is normal to the surface, and suppose the surface is represented as the graph of $z_0 + s(x - x_0, y - y_0)$ near $(x_0,y_0)$, with $s(0,0) = 0$. It is shown that as long as the Taylor series of the Hessian determinant of $s(x,y)$ at $(0,0)$ is not identically zero, the maximal averaging operator is bounded on $L^p$ for $p > \max(2,1/g)$, where $g$ is an index based on the growth rate of the distribution function $s(x,y)$ near the origin. Standard examples show that the exponent $1/g$ is best possible whenever the tangent plane to $S$ at $(x_0,y_0,z_0)$ does not contain the origin. This theorem improves on the main result of [IKeM], using different methods. We use closely related methods prove $L^p$ to $L^p_{\alpha}$ Sobolev estimates for Radon transform operators with the same density functions, with no excluded cases. In the $g < 1/2$ case, there is an interval $I$ containing $2$ for which $L^p$ to $L^p_{\alpha}$ boundedness is proven for $\alpha < g$ when $p \in I$, and for such $p$ one can never gain more than $g$ derivatives.