Rank and regularity for averages over submanifolds

TL;DR

This paper investigates the mapping properties of Radon-like operators with specific homogeneity and rank conditions, revealing their behavior in endpoint $L^p-L^q$ and Sobolev spaces, especially in degenerate cases.

Contribution

It establishes new endpoint $L^p-L^q$ and Sobolev estimates for Radon-like operators satisfying particular homogeneity and rank conditions, including highly degenerate cases.

Findings

Endpoint $L^p-L^q$ mapping properties are established.

Sobolev space mapping properties are characterized.

Degenerate operators satisfy rank conditions for algebraic reasons.

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.