Generalized curvature for certain Radon-like operators of intermediate dimension

TL;DR

This paper introduces a new notion of curvature to establish sharp $L^p$-improving estimates for Radon-like transforms over intermediate-dimensional submanifolds, extending previous methods without requiring specific linear algebraic conditions.

Contribution

It develops a novel curvature concept and adapts the inflation method to prove sharp $L^p$ bounds for a broad class of Radon-like operators of intermediate dimension.

Findings

Established sharp $L^p$-improving estimates for Radon-like transforms.

Introduced a new curvature notion related to but distinct from Phong-Stein curvature.

Extended the inflation method without requiring algebraic relations between dimensions.

Abstract

This paper establishes $L^p$-improving estimates for a variety of Radon-like transforms which integrate functions over submanifolds of intermediate dimension. In each case, the results rely on a unique notion of curvature which relates to, but is distinct from, Phong-Stein rotational curvature. The results obtained are sharp up to the loss of endpoints. The methods used are a new adaptation of the familiar method of inflation developed by Christ and others. Unlike most previous instances of this method, the present application does not require any particular linear algebraic relations to hold for the dimension and codimension.