On Sublevel Set Estimates and the Laplacian

Abstract

Carbery proved that if $u:\mathbb {R}^{n} \rightarrow \mathbb {R}$ is a positive, strictly convex function satisfying $\det D^{2}u \geq 1$, then we have the estimate

$$ \left| \left\{x \in \mathbb{R}^{n}: u(x) \leq s \right\} \right| \lesssim_{n} s^{n/2} $$

and this is optimal. We give a short proof that also implies other results. Our main result is an estimate for the sublevel set of functions $u:[0,1]^{2} \rightarrow \mathbb {R}$ satisfying 1 ≤Δu ≤ c for some universal constant c: for any α > 0, we have

$$ \left| \left\{x \in [0,1]^{2} : |u(x)| \leq \varepsilon\right\}\right| \lesssim_{c} \sqrt{\varepsilon} + \varepsilon^{\alpha - \frac12} {\int}_{[0,1]^{2}}{\frac{|\nabla u|}{|u|^{\alpha}} dx}.$$

For ‘typical’ functions, we expect the integral to be finite for α < 1. While Carbery-Christ-Wright have shown that no sublevel set estimates independently of u exist, this result shows that for ‘typical’ functions satisfying ${\Delta } u \sim 1$, we expect the sublevel set to be $\lesssim \varepsilon ^{1/2-}$. We do not know whether this is sharp or whether similar statements are true in higher dimensions.