  This paper gives a complete geometric characterization in all dimensions and
codimensions of those Radon-like transforms which, up to endpoints, satisfy the
largest possible range of local $L^p \rightarrow L^q$ inequalities permitted by
quadratic-type scaling. The necessary and sufficient curvature-type criterion
is phrased in terms of an associated Newton-like diagram. In the case of
averages over families of polynomial graphs, the curvature condition implies
sharp endpoint estimates as well. The proof relies on the recently-developed
multilinear Radon-Brascamp-Lieb testing criterion and a refined version of
differential inequalities for polynomials first appearing in work on the
Oberlin affine curvature condition.
