Level set estimates for strictly convex and hyperbolic functions

Abstract

In this note, we prove uniform upper bounds for the volume of the level set $$\{x\in \Omega :c\le f(x)<c+\delta \},c\in \mathbb{R},\delta >0,$$ for strictly convex and hyperbolic functions $f$ defined on a convex domain $\Omega \subset {\mathbb{R}}^n$ ($n\ge 2$). In particular, under a Hessian lower bound $D^{2}f\ge I_n$, we obtain the sharp volume bound $$|S^{n-1}|r(\Omega {)}^{n-2}\delta ,$$ where $r(\Omega )=\frac{1}{2}\mathrm{diam}\Omega$. As an application, we derive $L^2$ estimates for oscillatory integrals with convex phases.