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This paper gives a complete geometric characterization in all dimensions andcodimensions of those Radon-like transforms which, up to endpoints, satisfy thelargest possible range of local $L^p \rightarrow L^q$ inequalities permitted byquadratic-type scaling. The necessary and sufficient curvature-type criterionis phrased in terms of an associated Newton-like diagram. In the case ofaverages over families of polynomial graphs, the curvature condition impliessharp endpoint estimates as well. The proof relies on the recently-developedmultilinear Radon-Brascamp-Lieb testing criterion and a refined version ofdifferential inequalities for polynomials first appearing in work on theOberlin affine curvature condition.

Local curvature of maximally nondegenerate Radon-like transforms