Testing conditions for multilinear Radon-Brascamp-Lieb inequalities

TL;DR

This paper provides a comprehensive testing condition for the $L^p$-boundedness of multilinear Radon-Brascamp-Lieb inequalities, unifying several classical results and introducing new curvature and transversality insights.

Contribution

It establishes a necessary and sufficient testing condition for multilinear inequalities, extending the scope to generalized Radon transforms and revealing new interactions between curvature and transversality.

Findings

Derived a unified testing condition for $L^p$-boundedness.

Applied results to convolution with hypersurface measures on paraboloids.

Demonstrated new $L^p$-improving inequalities in specific geometric contexts.

Abstract

This paper establishes a necessary and sufficient condition for $L^p$-boundedness of a class of multilinear functionals which includes both the Brascamp-Lieb inequalities and generalized Radon transforms associated to algebraic incidence relations. The testing condition involves bounding the average of an inverse power of certain Jacobian-type quantities along fibers of associated projections and covers many widely-studied special cases, including convolution with measures on nondegenerate hypersurfaces or on nondegenerate curves. The heart of the proof is based on Guth's visibility lemma in one direction and on a careful analysis of Knapp-type examples in the other. Various applications are discussed which demonstrate new and subtle interplay between curvature and transversality and establish nontrivial mixed-norm $L^p$-improving inequalities in the model case of convolution with affine hypersurface measure on the paraboloid.