  $L^p$ to $L^p_{\beta}$ boundedness theorems are proven for translation
invariant averaging operators over hypersurfaces in Euclidean space. The
operators can either be Radon transforms or averaging operators with
multiparameter fractional integral kernel. In many cases, the amount $\beta > 0
$ of smoothing proven is optimal up to endpoints, and in such situations this
amount of smoothing can be computed explicitly through the use of appropriate
Newton polyhedra.
