2022/08/12 by Esmayli, Behnam, Ikonen, Toni, Rajala, Kai · 1 citation
#28A25 #28A75 #28A78 #30L10 #30L15 #FOS: Mathematics #Metric Geometry (math.MG)
paper · doi:10.48550/arxiv.2208.06185
We study coarea inequalities for metric surfaces -- metric spaces that are topological surfaces, without boundary, and which have locally finite Hausdorff 2-measure H2. For monotone Sobolev functions u\colon X → ℝ , we prove the inequality ∫ ℝ * ∫ u-1(t) g dH1 dt ≤ κ ∫ X g ρ dH2 \quadfor every Borel g \colon X → [0,∞], where ρ is any integrable upper gradient of u. If ρ is locally L2-integrable, we obtain the sharp constant κ=4/π. The monotonicity condition cannot be removed as we give an example of a metric surface X and a Lipschitz function u \colon X → ℝ for which the coarea inequality above fails.