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We prove $L^p$ boundedness results, $p > 2$, for local maximal averagingoperators over a smooth 2D hypersurface $S$ with either a $C^1$ densityfunction 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 thesurface 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 seriesof the Hessian determinant of $s(x,y)$ at $(0,0)$ is not identically zero, themaximal 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 $s(x,y)$ nearthe origin. Standard examples show that the exponent $1/g$ is best possiblewhenever the tangent plane to $S$ at $(x_0,y_0,z_0)$ does not contain theorigin. This theorem improves on the main result of [IKeM], using differentmethods.  We use closely related methods prove $L^p$ to $L^p_{\alpha}$ Sobolevestimates for Radon transform operators with the same density functions, withno 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.

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