A Nonlinear Variant of Ball's Inequality
Jennifer Duncan
TL;DR
The paper advances global nonlinear Brascamp--Lieb inequalities on manifolds with bounded geometry by proving a nonlinear Ball-type near-monotonicity under a geodesic heat-flow. It introduces a heat-flow operator $H_{x,\tau,j}$ and constructs near-extremising Gaussians $G_{x,\tau}$ via a regularised, differentiable Lieb framework, enabling a tight induction-on-scales analysis. A key contribution is a quantitative, smooth map $Y_{\delta}$ producing $\delta$-near extremisers with controlled norms, together with Gaussian and geometric lemmas that allow precise perturbations and localisation. The results yield explicit $\tau$-dependent error terms (via a $1+\tau^{\beta}$ factor) and show stability under bounded perturbations of the Brascamp--Lieb data, providing tools toward a general theory of global nonlinear Brascamp--Lieb inequalities. Overall, the work broadens the nonlinear BL toolkit beyond symmetric or homogeneous cases and connects semigroup methods with a robust geometric–analytic framework.
Abstract
We adapt an induction-on-scales argument of Bennett, Bez, Buschenhenke, Cowling, and Flock to establish a global near-monotonicity statement for the nonlinear Brascamp-Lieb functional under a certain heat-flow, from which follows a stability result for the finiteness of global nonlinear Brascamp-Lieb inequalities.