Coarea inequality for monotone functions on metric surfaces

Abstract

We study coarea inequalities for metric surfaces — metric spaces that are topological surfaces, without boundary, and which have locally finite Hausdorff 2-measure H 2 \mathcal {H}^2 . For monotone Sobolev functions u : X → R u\colon X \to \mathbb {R} , we prove the inequality ∫ R ∗ ∫ u − 1 ( t ) g d H 1 d t ≤ κ ∫ X g ρ d H 2 for every Borel g : X → [ 0 , ∞ ] , \begin{equation*} \int _{ \mathbb {R} }^{*} \int _{ u^{-1}(t) } g \,d\mathcal {H}^{1} \,dt \leq \kappa \int _{ X } g \rho \,d\mathcal {H}^{2} \quad \text {for every Borel $g \colon X \rightarrow \left [0,\infty \right ]$,} \end{equation*} where ρ \rho is any integrable upper gradient of u u . If ρ \rho is locally L 2 L^2 -integrable, we obtain the sharp constant κ = 4 / π \kappa =4/\pi . The monotonicity condition cannot be removed as we give an example of a metric surface X X and a Lipschitz function u : X → R u \colon X \to \mathbb {R} for which the coarea inequality above fails.