Rank and regularity for averages over submanifolds

Abstract

This paper establishes endpoint $L^p-L^q$ and Sobolev mapping properties of Radon-like operators which satisfy a homogeneity condition (similar to semiquasihomogeneity) and a condition on the rank of a matrix related to rotational curvature. For highly degenerate operators, the rank condition is generically satisfied for algebraic reasons, similar to an observation of Greenleaf, Pramanik, and Tang concerning oscillatory integral operators.