Rank and regularity for averages over submanifolds

This paper establishes endpoint Lp–Lq 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 [A. Greenleaf, M. Pramanik, W. Tang, Oscillatory integral operators with homogeneous polynomial phases in several variables, J. Funct. Anal. 244 (2) (2007) 444–487] concerning oscillatory integral operators. © 2009 Elsevier Inc. All rights reserved