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

TL;DR

This paper investigates the mapping properties of Radon-like operators in high dimensions, revealing that nonvanishing rotational curvature is rare and impossible for most dimensions, yet similar estimates are densely achievable.

Contribution

It demonstrates the rarity of nonvanishing rotational curvature in high-dimensional Radon-like operators and shows that operators with similar $L^p$-$L^q$ estimates are dense despite this rarity.

Findings

Nonvanishing rotational curvature is non-generic for $n \u2265 2$.

Such curvature is impossible for all but finitely many $n$.

Operators with similar $L^p$-$L^q$ estimates are dense in the family.

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 $n$.