2019/05/30 by Stefan Steinerberger, Steinerberger, Stefan
Mathematics · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA
paper · pdf · doi:10.48550/arxiv.1905.13176
arxiv created 2019/10/03 · arxiv updated 2019/10/04
Carbery proved that if u:ℝn → ℝ is a positive, strictly convex function satisfying det D2u ≥ 1, then we have the estimate | \x ∈ ℝn: u(x) ≤ s \ | \lesssimn sn/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 → ℝ satisfying 1 ≤ Δu ≤ c for some universal constant c: for any α> 0, we have | \x ∈ [0,1]2 : |u(x)| ≤ ε\| \lesssimc √(ε) + εα- \frac12 ∫[0,1]2(|∇ u|)/(|u|α) 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 independent of u exist, this result shows that for 'typical' functions satisfying Δu ≥ 1, we expect the sublevel set to be \lesssim ε1/2-. We do not know whether this is sharp or whether similar statements are true in higher dimensions.