$L^p$-nondegenerate Radon-like operators with vanishing rotational curvature

Abstract:

We consider the $L^p \rightarrow L^q$ mapping properties of a model family of Radon-like operators integrating functions over $n$-dimensional submanifolds of $\mathbb {R}^{2n}$. It is shown that nonvanishing rotational curvature is never generic when $n \geq 2$ and is, in fact, impossible for all but finitely many values of $n$. Nevertheless, operators satisfying the same $L^p \rightarrow L^q$ estimates as the “nondegenerate” case (modulo the endpoint) are dense in the model family for all $n$.