Year: 2026
Type: preprint
Abstract: In this note, we prove uniform upper bounds for the volume of the level set $$\{x\inΩ: c\le f(x)0,$$ for strictly convex and hyperbolic functions $f$ defined on a convex domain $Ω\subset\mathbb{R}^n$ ($n\ge 2$). In particular, under a Hessian lower bound $D^2f\ge I_n$, we obtain the sharp volume bound $$|S^{n-1}|\, r(Ω)^{n-2}\,δ,$$ where $r(Ω)=\frac12\,{\mathrm{diam}Ω}$. As an application, we derive $L^2$ estimates for oscillatory integrals with convex phases.
Source: arXiv (Cornell University)
Authors: Chen, Xianghong, Yang, Tongou
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214737729