Uniform sublevel Radon-like inequalities

TL;DR

This paper develops uniform weighted $L^p$-$L^q$ estimates for Radon-like and sublevel set operators, introducing new affine-invariant and $L^p$-improving bounds, especially for degenerate cases with curvature conditions.

Contribution

It establishes novel weighted affine-invariant estimates and $L^p$-improving bounds for Radon-like operators, extending the scope beyond existing Fourier integral operator results.

Findings

New weighted affine-invariant estimates for Radon-like operators

Improved $L^p$-bounds for degenerate Radon-like operators with curvature conditions

Estimates outside the known Fourier integral operator range

Abstract

This paper is concerned with establishing uniform weighted $L^p$-$L^q$ estimates for a class of operators generalizing both Radon-like operators and sublevel set operators. Such estimates are shown to hold under general circumstances whenever a scalar inequality holds for certain associated measures (the inequality is of the sort studied by Oberlin, relating measures of parallelepipeds to powers of their Euclidean volumes). These ideas lead to previously unknown, weighted affine-invariant estimates for Radon-like operators as well as new $L^p$-improving estimates for degenerate Radon-like operators with folding canonical relations which satisfy an additional curvature condition of Greenleaf and Seeger for FIOs (building on the ideas of Sogge and Mockenhaupt, Seeger, and Sogge); these new estimates fall outside the range of estimates which are known to hold in the generality of the FIO context.