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 $$\{x\in\Omega: c\le f(x)0,$$ 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 $D^2f\ge I_n$, we obtain the sharp volume bound $$|S^{n-1}|\, r(\Omega)^{n-2}\,\delta,$$ where $r(\Omega)=\frac12\,{\mathrm{diam}\Omega}$. As an application, we derive $L^2$ estimates for oscillatory integrals with convex phases.