Coarea inequality for monotone functions on metric surfaces

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY-NC-ND 4.0 https://creativecommons.org/licenses/by-nc-nd/4.0/ Coarea inequality for monotone functions on metric surfaces © Authors 2023 Accepted version (Final draft) Esmayli, Behnam; Ikonen, Toni; Rajala, Kai Esmayli, B., Ikonen, T., & Rajala, K. (2023). Coarea inequality for monotone functions on metric surfaces. Transactions of the American Mathematical Society, 376, 7377-7406. https://doi.org/10.1090/tran/8998 2023 COAREA INEQUALITY FOR MONOTONE FUNCTIONS ON METRIC SURFACES BEHNAM ESMAYLI, TONI IKONEN, KAI RAJALA Abstract. 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:X→R, we prove the inequality ∗ ˆ R ˆ u−1(t) g dH1dt ≤κˆ X gρ dH2for every Borel g:X→[0,∞], where ρis any integrable upper gradient of u. If ρis 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 Xand a Lipschitz function u:X→Rfor which the coarea inequality above fails. 1. Introduction In this paper we prove a coarea inequality, involving upper gradients, for monotone Sobolev functions on metric surfaces. To motivate the topic, recall the classical coarea formula for Lipschitz maps u:Rn→Rm,n≥m≥1. When m= 1, the only relevant case in this paper, it reads as follows. Theorem 1.1. If Ω⊂Rnis open and u: Ω →Ris Lipschitz, then ˆ R ˆ u−1(t) g dHn−1dt =ˆ Ω g|∇u|dx for every Borel g: Ω →[0,∞].(1) Throughout, Hαstands for Hausdorff measures. Extension of this result to Sobolev class is a delicate matter, despite well-known Lipschitz approximation results. Theorem 1.2 (Coarea formula, [MSZ03]).Let Ω⊂Rnbe open and u∈W1,1 loc (Ω; R)be precisely represented. Then the coarea formula (1)holds with ∇ubeing the weak derivative. We recall that continuous functions are precisely represented. There are several equivalent approaches to Sobolev functions on metric(-measure) spaces. We use the definition based on upper gradients in the sense of Heinonen and Koskela [HK98,Sha00]. An upper gradient of a function u:X→R, on a metric space (X, d), is a Research supported by the Academy of Finland, project number 308659. The second named author was also supported by the Vilho, Yrjö and Kalle Väisälä Foundation. 2020 Mathematics Subject Classification. 28A25, 28A75, 28A78, 30L10, 30L15. 1 2 BEHNAM ESMAYLI, TONI IKONEN, KAI RAJALA Borel function ρ:X→[0,∞]so that for all x, y ∈Xand all rectifiable paths γjoining x and yin Xwe have |u(x)−u(y)| ≤ ˆ γ ρ ds. Observe that for C1-smooth functions on Euclidean domains, ρ(x) = |∇u(x)|meets this criterion. The upper gradient approach is equivalent to the energy density approach [AGS13,ES21] and the test plan approach [AGS13]. We refer the interested reader to [ACDM15]. On Euclidean domains, the upper gradient approach leads to the classical Sobolev theory, see e.g. [HKST15]. We restrict ourselves to the class of metric n-manifolds. By a metric n-manifold (resp. with boundary) we mean a metric space homeomorphic to a topological n-manifold (resp. with boundary) with locally finite n-dimensional Hausdorff measure. Unless otherwise mentioned, a metric n-manifold is assumed to have an empty boundary. When n= 2, and there is no boundary, we use the term metric surface. Question 1.3. (Coarea inequality) Does there exist a universal constant C=C(n)such that for all metric n-manifolds X, all u:X→Rand any upper gradient ρ:X→[0,∞]of u, with locally integrable uand ρ, ∗ ˆ R ˆ u−1(t) g dHn−1dt ≤Cˆ X gρ dHnfor every Borel g:X→[0,∞]? (2) Here ´∗ Rrefers to the upper integral in case the integrand happens to be nonmeasurable. Question 1.3 has a positive answer for Lipschitz functions in all metric n-manifolds that support a (1,1)-Poincaré inequality and on which Hnis doubling, cf. Section 5. In recent years the research on metric n-manifolds has been active, mainly for n= 2, where inequalities of the form (2) have played a prominent role in the uniformization results of metric surfaces by the third named author [Raj17] and more recently by Ntalampekos and Romney [NR22], see also [LW17,MW21]. The particular formulation of (2) is motivated by a related well-known inequality from geometric measure theory. Theorem 1.4 (Eilenberg’s Inequality, [Fed69,EH21])).Let (X, d)be any metric space and fix n≥1, not necessarily an integer, and suppose that Xhas a locally finite Hn-measure. Then for any Lipschitz function u:X→R, ∗ ˆ R ˆ u−1(t) g dHn−1dt ≤2ωn−1 ωnˆ X glip (u)dHn,for every Borel g:X→[0,∞],(3) where lip (u)(x):= lim sup x=y→x |u(y)−u(x)| d(y, x).(4) Here ωiare normalization constants involved in the definition of the Hausdorff measure. In particular, when nis an integer, ωnis the volume of the n-dimensional Euclidean unit ball. For example, ω1= 2 and ω2=π, so 2ω1/ω2= 4/π. COAREA INEQUALITY FOR MONOTONE FUNCTIONS ON METRIC SURFACES 3 Eilenberg’s inequality is often stated using the global Lipschitz constant instead of lip , but a localization argument leads to (3). See Section 5, in particular Lemma 5.2, for a proof of this folklore result. The Eilenberg inequality is closely related to the perimeter of sublevel sets of Lipschitz functions, cf. [Mir03]. However, without strong geometric assumptions on X, see e.g. [Amb01], the connection between the Hausdorff measure and perimeter measure is unclear. Theorem 1.4 implies, indirectly, that the Sobolev theory on metric surfaces is rich, cf. Section 1.1. In particular, it is possible to construct 2-harmonic functions for suitable boundary value problems on Xas was done by the third named author in [Raj17, Section 3]. The results therein generalize to p∈(1,∞)and also to related problems in potential analysis. We answer Question 1.3 in two-dimensions for a class of functions that contains such functions and other functions arising from energy minimization problems. First, a definition. Definition 1.5. A function u:X→Ris monotone if uis continuous and satisfies the maximum principle: for every open set U, compactly contained in X, sup U u= sup ∂U uand inf U u= inf ∂U u. Theorem 1.6. Let Xbe a metric surface and ∞ ≥ p≥1. If u:X→Ris a monotone function with a locally p-integrable upper gradient ρ, then for κ= (4/π)·200, ˆ R ˆ u−1(t) g dH1dt ≤κˆ X gρ dH2(5) for every Borel function g:X→[0,∞]. If p≥2, then (5)holds with constant κ= 4/π. We expect that the sharp constant κ= 4/π holds also for p < 2. Endowing the Euclidean plane with the supremum norm ∥(x1, x2)∥:= sup {|x1|,|x2|} and considering u(x1, x2) = x1 shows the sharpness of Theorem 1.6 when p≥2. As hinted at earlier, (3) implies (2) for Lipschitz functions in spaces satisfying strong geometric assumptions, for example, Hnbeing doubling and supporting a (1,1)-Poincaré inequality, cf. [Che99] (or [IPS22,ES21]). Without the further geometric assumptions, the pointwise Lipschitz constant may be much larger than the minimal upper gradient, so (3) does not always imply (2). Indeed, we have the following consequence of Theorem 5.3. Theorem 1.7. For every n≥2, there exist a metric n-manifold X⊂Rn+1 and a Cantor set C⊂Xsuch that for u(x1, x2, . . . , xn) = x1, the following holds: 0<Hn(C) = ˆ R Hn−1(C∩u−1(t)) dt and ρ=χX\Cis an upper gradient of u|X. In particular, (2)fails for g=χCwith any constant. Theorem 1.7 illustrates that the inequality (2) does not hold in the Lipschitz class. For this reason, there is no obvious generalization of Theorem 1.6 in dimension two. Question 1.8. Does Question 1.3 have a positive answer on metric n-manifolds for monotone functions u:X→Rwith (locally) integrable upper gradients? 4 BEHNAM ESMAYLI, TONI IKONEN, KAI RAJALA 1.1. Two-dimensionality. A noticeable assumption in Theorem 1.6 is the setting of metric surfaces. Our methods do not lend themselves to obvious generalization even to metric n-manifold setting for n≥3. To illustrate the point, we argue as follows. When Theorem 1.4 is applied on a metric n-manifold and u(x) = d(x, x0), we conclude that almost every level set of usatisfies Hn−1(u−1(t)) <∞. When tis small enough, u−1(t)is contained in a neighbourhood Ωof x0homeomorphic to Rn, and Alexander’s duality guarantees the existence of a continuum C⊂u−1(t)separating Ωinto two or more connected components. In the particular case of n= 2,H1(C)<∞guarantees that Cis a rectifiable path which allows us to control the oscillation of a Sobolev function fon Cin terms of ´Cρ dH1, where ρis any upper gradient of f. More precisely, we have |f(x)−f(y)| ≤ ˆ C ρ dH1for every x, y ∈C. (6) Without further geometric assumption on X, (6) does not seem to have a straight-forward generalization for n > 2since, e.g., Hn−1(C)<∞for continua Cdoes not imply the existence of a Lipschitz parametrization from [0,1]n−1. The key point is that (6) allows one to deduce that Sobolev analysis on metric surfaces is rich without any need for further assumptions on X. To handle the full range 1≤p≤ ∞, we provide two proofs of Theorem 1.6. The inequality involving the non-sharp constant holds in the full range but the sharp constant for p≥2in Theorem 1.6 is based on recent advances in uniformization theory of metric surfaces: the existence of (weakly) quasiconformal homeomorphisms onto metric surfaces [NR22], see Section 4.1 for further discussion. 1.2. Monotone functions. Typical examples of monotone functions include solutions to the p-Laplace equation or other elliptic PDE’s of divergence form. The calculus of variations approach to p-harmonic functions can be developed in metric (measure) spaces, cf. [BB11]. In fact, an important first step in the uniformization theorem by the third named author in [Raj17] was to construct a specific 2-harmonic function on a general metric surface. The constructed function satisfies the following slightly weaker property. Definition 1.9. A function u:X→Ris weakly monotone if for every open Vcompactly contained in X, sup V u≤sup ∂V u < ∞and inf Vu≥inf ∂V u > −∞. Deducing continuity of weakly monotone functions is of interest. This is one of our main results. Theorem 1.10. Suppose that Xis a metric surface and u:X→Ris weakly monotone. If uhas a locally p-integrable upper gradient, for p≥2, then uis continuous. In particular, uis monotone. The key idea in the proof of Theorem 1.10 is a non-sharp version of the coarea inequality for weakly monotone functions with locally integrable upper gradients. Our approach is based on a related result [RR19], and in fact, Theorem 1.10 generalizes related continuity results from [RR19] for all ranges p≥2and for a large class of problems. Moreover, COAREA INEQUALITY FOR MONOTONE FUNCTIONS ON METRIC SURFACES 5 together with Theorem 1.6, we obtain a sharp duality of modulus lower bound in [RR19, Theorem 1.3]; we note that the sharp duality lower bound was first proved in [EBPC22, Corollary 1.2] with different methods and in greater generality. When p < 2, the continuity conclusion does not hold even in the plane, cf. Example 3.10. Given an inequality of the form (5) for monotone functions, a consequence of [Nta20, Theorem 1.5] follows for metric surfaces. Corollary 1.11. Let Ube a metric surface homeomorphic to R2, and p≥1. If a monotone u:U→Rhas a locally p-integrable upper gradient, then for almost every t∈u(U)the following properties hold: (a) The level set u−1(t)is a locally rectifiable properly embedded topological 1-manifold. (b) Each component of u−1(t)is homeomorphic to R. We say that a set K⊂Uis a properly embedded topological 1-manifold if Kis closed and every y∈Kis contained in I⊂K, relatively open in K, with Ihomeomorphic to R. Corollary 1.11 plays a key role when we establish the sharp version of (5) for p≥2. Notation. We shall write u−1(t)to mean {x:u(x) = t}. The α-dimensional Hausdorff measure is denoted by Hα. The upper integral of any function on a measure space (X, µ) is denoted by ´∗f dµ. If fis µ-measurable then it agrees with the usual integral. We use #A,χAand Ato denote, resp., the cardinality, the characteristic function, and closure of a set A. The closed ball {y:d(y, x)≤r}is denoted by B(x, r), which might not coincide with the closure of the open ball B(x, r) = {y:d(y, x)< r}. 2. Preliminaries Let Xbe a metric space. For all Q≥0, the Q-dimensional Hausdorff measure, or the Hausdorff Q-measure, of a set E⊂Xis defined by HQ X(E) = ωQ 2Qsup δ>0 inf (∞ X i=1 (diam Ei)Q:E⊂ ∞ [ i=1 Ei,diam Ei< δ), where the dimensional constant ωQis chosen so that Hn Rncoincides with the Lebesgue measure Lnfor all positive integers. In particular, ω1= 2 and ω2=π. We typically omit the subscript Xfrom the definition. Given a set K⊂X, a function f:K→Ris Lipschitz if LIP(f) := sup x,y∈K,x=y |f(x)−f(y)| d(x, y)<∞. The supremum on the left is the Lipschitz constant of f. We say that fis L-Lipschitz if LIP(f)≤L. For a given Lipschitz f:K→Rand x∈K, we define the pointwise Lipschitz constant of fas lip (f)(x) = inf r>0sup 0<s≤r sup y∈B(x,s)∩K |f(y)−f(x)| s. 6 BEHNAM ESMAYLI, TONI IKONEN, KAI RAJALA In fact, this definition of lip (f)coincides with the one in (4). Let Xbe a metric surface. For each 1≤p < ∞, we say ρ:X→[−∞,∞]belongs to Lp(X)if ρis measurable and ∥ρ∥Lp(X):= ˆ X |ρ|pdH2  1/p <∞. For p=∞,∥ρ∥L∞(X)is the smallest C∈[0,∞]for which |ρ| ≤ C,H2-almost everywhere, and denote ρ∈L∞(X)if ρis measurable and ∥ρ∥L∞(X)<∞. In case ρ∈Lp(X), we say that ρis p-integrable. Local p-integrability refers to being p-integrable on each compact subset of the space; recall that Xis locally compact so the space Xcan be covered by open sets whose closures are compact, justifying the nomenclature. 2.1. The upper integral. Let E⊂Xbe a set with HQ(E)<∞for Q= 2 (resp. Q= 1). For any function ρ:E→[0,∞]we define the upper integral of ρ(with respect to HQ) to be ∗ ˆ E ρ dHQ:= inf   ˆ E ρ′dHQ:ρ′is HQ-measurable and ρ≤ρ′,HQ-almost everywhere  . We use some elementary properties of the upper integral. If 0≤ρ1(x)≤ρ2(x),HQalmost everywhere in E, then ∗ ˆ E ρ1dHQ≤ ∗ ˆ E ρ2dHQ. The monotone convergence theorem holds for upper integrals. Namely, if 0≤ρ1(x)≤ ρ2(x)≤. . . is an increasing sequence of (not necessarily measurable) functions, and for HQ-almost every x∈E,ρ(x) = limn→∞ ρn(x), then ∗ ˆ E ρ dHQ= lim n→∞ ∗ ˆ E ρndHQ. Lastly, for an arbitrary ρ:E→[0,∞],´∗ Eρ dHQ= 0,if and only if ρ= 0,HQ-almost everywhere in E. 2.2. Rectifiable curves and path integrals. Apath in a metric space Xis a continuous map γ: [a, b]→X. The length ℓ(γ)of γis the smallest value L∈[0,∞]for which k X i=1 d(γ(ti), γ(ti−1)) ≤L, for every choice of k∈Nand a=t0≤t1≤ · · · ≤ tk=b. We say that γis rectifiable if ℓ(γ)<∞. COAREA INEQUALITY FOR MONOTONE FUNCTIONS ON METRIC SURFACES 7 Suppose γ: [a, b]→Xis a path and ρ:X→[0,∞]is Borel. Then the path integral of ρover γis ˆ γ ρ ds :=ˆ X #(γ−1(x))ρ(x)dH1. Here #(γ−1(x)) = ∞if γ−1(x)is not finite and otherwise #(γ−1(x)) is the cardinality of the set γ−1(x). If ρis not Borel, we define ˆ γ ρ ds := ∗ ˆ X #(γ−1(x))ρ(x)dH1. Remark 2.1. Note that whenever E⊂[a, b]is Borel and γ: [a, b]→Xis a path, then γ(E)is analytic [Fed69, 2.2.10]. This implies that γ(E)is H1-measurable [Fed69, 2.2.13]. This allows us to prove that x7→ #(γ−1(x)) is H1-measurable. The key observation is to fix a sequence of countable Borel partitions (Kn)of [a, b]such that the supremum of the diameters of the elements of Knconverges to zero as n→ ∞ and each Kn+1 refines Kn, i.e., each E∈ Knis a countable union of some elements of Kn+1. Now, measurability follows from #(γ−1(x)) = lim n→∞ X E∈Kn χγ(E)(x)for every x∈X. If γ: [a, b]→Xis rectifiable, then there exists a unique path γs: [0, ℓ(γ)] →Xsuch that γ=γs◦h, where h: [a, b]→[0, ℓ(γ)] is continuous, nondecreasing and onto, and ℓ(γs|[0,s]) = sfor all 0≤s≤ℓ(γ). The path γsis called the arclength parametrization of γ. Recall that the arclength parametrization is 1-Lipschitz, cf. [HKST15, Section 5]. Let γ: [a, b]→Xbe a rectifiable path in a metric space, and let γs: [0, ℓ(γ)] →Xbe the arclength parametrization of it. For a Borel function ρ:X→[0,+∞], the path integral of ρover γcan be computed as follows: ˆ γ ρ ds = ℓ(γ) ˆ 0 ρ(γs(t)) dt. The equality follows from the area formula for paths, proved for example in [Fed69, Theorem 2.10.13.]. A path γ: [a, b]→Xis absolutely continuous if ℓ(γ)<∞and if γmaps sets of Lebesgue measure zero to sets of H1-measure zero. For absolutely continuous curves, there is a third way to compute the path integral of Borel functions. For this purpose, we denote |γ′|(t) := lim h→0 d(γ(t+h), γ(t)) |h| whenever the limit exists. When the limit exists, we refer to |γ′|(t)as the metric speed of γat t. It turns out that for any rectifiable path, the limit exists almost everywhere in [a, b][Dud07]. Recall that in the Euclidean setting, the metric speed coincides with the modulus of the usual derivative. With the additional assumption of absolute continuity, we obtain the following. 8 BEHNAM ESMAYLI, TONI IKONEN, KAI RAJALA Lemma 2.2 ([Dud07]).Suppose that γ: [a, b]→Xis absolutely continuous and ρ:X→ [0,+∞]is Borel. Then ˆ γ ρ ds = b ˆ a ρ(γ(t))|γ′|(t)dt. 2.3. Modulus of path families. We continue considering a metric surface X. We typically denote a collection of paths by Γand refer to Γas a path family. A Borel function ρ:X→[0,∞]is admissible for a path family Γif ˆ γ ρ ds ≥1for every γ∈Γ. Then, for each 1≤p < ∞, we denote modpΓ = inf   ˆ X ρpdH2 X:ρis admissible for Γ  . In case p=∞, we set modpΓ = inf ∥ρ∥L∞(X):ρis admissible for Γ. The set function Γ7→ modpΓis an outer measure. We say that Γis p-negligible if modpΓ=0. The following characterization of negligible paths is an effective tool. Notice that this characterization does not require the notion of modulus and could be given as a definition of modulus zero without defining modulus, see, e.g. [HKST15, Lemma 5.2.8] for a proof. Lemma 2.3. Let 1≤p≤ ∞. A path family Γis p-negligible if and only if there exists an Lp(X)-integrable Borel function h:X→[0,∞]such that ´γh ds =∞for every γ∈Γ. 2.4. Sobolev analysis. Let Xbe a metric surface and Ya metric space. Let u:X→Y be a map and ρ:X→[0,∞]a Borel function. If γ: [a, b]→Xis rectifiable, we say that the triple (u, ρ, γ)satisfies the upper gradient inequality if d(u(γ(a)), u(γ(b))) ≤ˆ γ ρ ds. If the triple (u, ρ, γ)satisfies the upper gradient inequality for every path outside a pnegligible family, we say that ρis a p-weak upper gradient of u. If the exceptional set of paths is empty, we say ρis an upper gradient of u. If uhas a p-integrable p-weak upper gradient, then there exists a p-weak upper gradient ρsuch that ρ≤ρ′almost everywhere for every other p-integrable p-weak upper gradient ρ′ of u. For 1≤p < ∞, this is proved in [HKST15, Theorem 6.3.20] and for p=∞a similar argument works, cf. [Mal13]. Any p-minimal p-weak upper gradient of uis denoted by ρu; we typically omit the pfrom the notation since pis clear from the context. Whenever 1≤p≤ ∞, we write u∈D1,p(X;Y)whenever uhas a p-integrable p-weak upper gradient. In case Y=R, we also use u∈D1,p(X). We note that Lemma 2.3 COAREA INEQUALITY FOR MONOTONE FUNCTIONS ON METRIC SURFACES 15 Proof. We apply Proposition 3.1 for E={x0}and f(z) = d(z, E). Then (7) shows that 2 sup x,y∈V |u(y)−u(x)| ≤ ˆ f−1(s) ρ dH1for almost every s∈(r, 2r). Now, (13) follows by integrating over the interval (r, 2r)and applying the Eilenberg inequality. □ We later show continuity for weakly monotone functions with p-integrable upper gradients, p≥2. The following lemma is a weaker continuity result. Lemma 3.5. Let Gdenote the collection of all x∈Uwith lim sup r→0+ r−2ˆ B(x,2r) ρ dH2<∞. Then uis continuous at every x∈G. In particular, uis continuous H2-almost everywhere in U. To emphasize, here continuity is with respect to the original domain and not just the continuity of u|G. Proof. The complement of Ghas negligible H2-measure due to the integrability of ρ; we apply [Fed69, Theorem 2.10.19 (3)] to the measure µ=ρH2to deduce that the 2-dimensional upper density of µis finite at H2-almost every x∈X. Consider an arbitrary x0∈G. Since the collection of the sets Vfrom Corollary 3.4 form a neighbourhood basis of x0, continuity of uat x0follows from (13) and the defining property of G.□ 3.2. First proof of the coarea inequality. In this section we prove Theorem 1.6 for weakly monotone functions and weak upper gradients, with the non-sharp constant κ. Theorem 3.6. Let Xbe a metric surface and p≥1. If u:X→Ris a weakly monotone function with a locally p-integrable p-weak upper gradient ρ, then for κ= (4/π)·200, ∗ ˆ R ˆ u−1(t) g dH1dt ≤κˆ X gρ dH2for every Borel g:X→[0,∞].(14) Proof. It suffices to show that ∗ ˆ R ˆ u−1(t)∩U g dH1dt ≤κˆ U gρ dH2for every Borel g:U→[0,∞](15) for any subset U⊂Xhomeomorphic to R2with H2(U)<∞and g∈Lp(U). Indeed, we can cover Xwith countably many such subsets Uj, apply (15) to the restrictions of gto Uj\ ∪j−1 ℓ=1Uℓon Uj, and sum up the results to get (14). 16 BEHNAM ESMAYLI, TONI IKONEN, KAI RAJALA Moreover, it suffices to prove (15) for g=χE, the characteristic function of an open set E⊂Ucompactly contained in U, due to standard approximation via simple functions. Fix ϵ0>0such that B(x, 20ϵ0)is compactly contained in Ufor every x∈E. Step 1: Consider the collection Gof all x∈Efor which for all 0< ϵ < ϵ0there exists some 0< r < ϵ with ˆ B(x,10r) ρ dH2≤200 ˆ B(x,r) ρ dH2. Fix an arbitrary 0< ϵ < ϵ0. By the 5r-covering theorem [Fed69, 2.8.4], there exists a countable (possibly finite) collection of pairwise disjoint balls {B(xj, rj)}j, for which xj∈G,10rj<min {ϵ, d(X\E, xj)}. We write Bj=B(xj, rj)for short. The collection {B(xj,5rj)}jcovers G, and ˆ 10Bj ρ dH2≤200 ˆ Bj ρ dH2for each j. For each j, we find a Borel set Cj⊃u(5Bj)with H1(Cj) = H1(u(5Bj)). Then gϵ(t) = X j 10rjχCj(t) is Borel measurable. Then, by Corollary 3.2, applied with r= 5rj, ˆ R gϵ(t)dt ≤X j 4 πˆ 10Bj ρ dH2. The defining property of the Bjand the inclusion SjBj⊂Eyield X jˆ 10Bj ρ dH2≤200 ˆ E ρ dH2. Thus, ˆ R gϵ(t)dt ≤4 π200 ˆ E ρ dH2. Suppose that x∈u−1(t)∩Gfor some given t∈R. Then xis contained in some 5Bj. By openness of 5Bjone shows that t∈u(5Bj)for every such j. Hence the definition of the Hausdorff content H1 ϵyields H1 ϵ(u−1(t)∩G)≤X j:t∈u(5Bj) 10rj≤X j 10rjχu(5Bj)(t)≤gϵ(t). Since ϵwas arbitrary, by applying monotone convergence theorem, we conclude ∗ ˆ R H1(u−1(t)∩G)dt = lim ϵ→0+ ∗ ˆ R H1 ϵ(u−1(t)∩G)dt ≤4 π200 ˆ E ρ dH2. COAREA INEQUALITY FOR MONOTONE FUNCTIONS ON METRIC SURFACES 17 Step 2: Consider F=E\G. We claim that ∗ ˆ R H1(u−1(t)∩F)dt = 0.(16) This will complete the proof of inequality (15) for g=χE. Indeed, having verified (16), by Step 1 and subadditivity property of the upper integral we deduce ∗ ˆ R H1(u−1(t)) dt = ∗ ˆ R H1(u−1(t)∩G)dt ≤4 π200 ˆ E ρ dH2. So it remains to establish (16). By definition of G, for every x∈F, there exists kx∈N, such that for any j > kx, ˆ B(x,10−j) ρ dH2≤200−(j−kx)ˆ B(x,10−kx) ρ dH2.(17) By monotone convergence, it is sufficient to establish (16) with Freplaced with Fk=x∈F:kx≤k, d(x, X \E)>10−kfor arbitrary k∈N. We fix k∈Nand j−1> k for now. By the definition of the Hausdorff measure, there exists a countable collection of balls Bmthat cover Fk, with radii rmthat satisfy rm≤10−j and 2rm≥diam Bm≥rm, such that each Bmintersects Fk, and 4 πX m (diam Bm)2−(1/j)≤4H2(Fk).(18) In fact, we may require the balls to be centered at the set Fk.1 For each m∈N, let jm∈Zbe the largest integer for which 2rm≤10−jm. Observe from the inequalities 10−jm<20rm≤20 ·10−jthat jm≥j−1. Using these observations, we deduce from Corollary 3.2 and (17) that 2rmH1(u(Bm)) ≤4 πˆ 2Bm ρ dH2≤4 π200−(jm−k)ˆ U ρ dH2.(19) As before, we consider gj(x) = Pm2rmχCmfor Borel sets Cm⊃u(Bm)with H1(Cm) = H1(u(Bm)). By arguing as in Step (1), the definition of H1 1/j yields that H1 1/j(u−1(t)∩Fk)≤gj(t)for all t∈R. By (upper) integrating over R, we obtain ∗ ˆ R H1 1/j(u−1(t)∩Fk)dt ≤X m 2rmH1(u(Bm)). 1After an initial choice of a covering according to the definition of H2, replace each set by a closed ball centered on Fkand radius equal to the diameter of the set. (Hence, the factor 4on the right.) 18 BEHNAM ESMAYLI, TONI IKONEN, KAI RAJALA We apply now (19) and the inequalities 1<20 ·10jm·rmand jm≥j−1, and obtain ∗ ˆ R H1 1/j(u−1(t)∩Fk)dt ≤X m 2rmH1(u(Bm)) ≤4 πX m 200−(jm−k)ˆ U ρ dH2 ≤200k·4 π ˆ U ρ dH2 X m 200−jm(20 ·10jmrm)2 ≤200k·4 π ˆ U ρ dH2 202X m 2−(j−1)r2 m ≤200k·4 π ˆ U ρ dH2 2022−(j−1) X m r2 m. Now, we apply (18) and pass to the limit as j→ ∞, and conclude ∗ ˆ R H1(u−1(t)∩Fk)dt = 0. Passing to the limit k→ ∞ yields (16) and the proof is complete. □ 3.3. Continuity of weakly monotone functions. In this section we prove Theorem 1.10: weakly monotone functions with p-integrable upper gradients are continuous when p≥2. In other words, for this range of p,weakly monotone functions are monotone functions. We prove this result by a refined study of the topology of the level sets of such functions. The standing assumptions in this section are that Uis homeomorphic to R2with H2(U)<∞, and u:U→Ris weakly monotone (Definition 1.9). We moreover assume that uhas a p-integrable upper gradient ρ,1≤p < ∞. We start with the following topological lemma, cf. [Nta20, Corollary 2.8], which says that connected components of the closures of the level sets of weakly monotone functions “leave every compact set”. Proposition 3.7. Let E⊂u−1(t)∩Ube a connected component. Then ∅ =E\Kfor every compact K⊂U. Remark 3.8. Since the results are applied to cases where Uis a subset of a metric surface X, we maintain the notation u−1(t)∩Uto emphasize that we are taking the closure relative to the subspace topology of U. Proof. Aiming for a contradiction, suppose to the contrary that E\K=∅for some compact K⊂U. Then Eitself is compact. Consider then a Jordan domain W⊃Ecompactly contained in U. We denote A:=W∩u−1(t), and observe that Eis also a connected component of A. COAREA INEQUALITY FOR MONOTONE FUNCTIONS ON METRIC SURFACES 19 Since Ais a compact subset of U, with Ubeing homeomorphic to R2, for each open set V⊃E, there exists a Jordan domain V′compactly contained in V, with E⊂V′and A∩∂V ′=∅; see [Why58, Corollary 3.11, p.35]. Below, we apply this result for V=W. We apply Lemmas 2.5 and 2.13 to f(z) = d(z, ∂V ′)as follows: Let ϵ0= min d(U\W, ∂V ′), d(A, ∂V ′), observing that f−1([0, s]) is compact in W\Afor every 0< s < ϵ0. Then, for every 0< s < ϵ0, there exists a connected component Fs⊂f−1(s)∩V′separating Efrom ∂V ′; recall Lemma 2.5. By applying Lemma 2.13, for almost every such s, we may assume that Fsis simple (thus, homeomorphic to S1, cf. Lemma 2.7) and admits a Lipschitz parametrization along which uis absolutely continuous. We fix one such sand let Vsbe the Jordan domain bounded by Fsthat contains E. Since E⊂Vs∩u−1(t), Definition 1.9 implies t∈inf ∂Vs u, sup ∂Vs u. Given that u|∂Vsis continuous, there exists x0∈∂Vssuch that u(x0) = t. But then, x0∈u−1(t)∩∂Vs, contradicting A∩∂Vs=∅.□ The next result can be compared to Lemma 2.13. However, here uis not necessarily Lipschitz, and instead of the Eilenberg inequality we use the coarea inequality of Theorem 3.6. Recall that a simple Peano continuum is a set homeomorphic either to a point, or a compact interval, or to S1. Proposition 3.9. Let u:U→Rbe a weakly monotone function with a p-integrable p-weak upper gradient, for some 2≤p≤ ∞. Denote I:= u(U). Then, for almost every t∈I, (1) u−1(t) = u−1(t)∩Uand every continuum E⊂u−1(t)∩Uis a simple Peano continuum, and, (2) H1(u−1(t)∩U)<∞. Suppose, moreover, that Γ0is a path family with p-modulus zero and N0⊂Uis an H2negligible set. Then, for almost every t∈I, (3) for every continuum E⊂u−1(t)∩Uof positive diameter, there exists a surjective Lipschitz path γ: [0,1] →Ethat is injective outside its end points; and (4) for every absolutely continuous γ: [0,1] →u−1(t)∩U,´γχN0ds = 0. Moreover, if there exists M∈Nsuch that #(γ−1(x)) ≤Mfor H1-almost every x∈u−1(t)∩U, then γ∈ Γ0. Proof. We fix a minimal p-weak upper gradient ρof u. Let Γ1denote the negligible collection of rectifiable paths for which the triple (u, ρ, γ)fails the upper gradient inequality or along which ρfails to be path integrable. Recall the negligible family Γ0and the set N0with H2(N0)=0from our assumptions, and recall also that uis continuous outside some N1with H2(N1)=0, as stated in 20 BEHNAM ESMAYLI, TONI IKONEN, KAI RAJALA Lemma 3.5. Fix an arbitrary Borel set B⊃N0∪N1with H2(B)=0. We consider a Borel function G:U→[0,∞]in Lp(U)satisfying ˆ γ G ds =∞,for all γ∈Γ0∪Γ1. Let bρ:=ρ+ (1 + G)ϵ+∞ · χBfor an arbitrary ϵ > 0. Observe that bρis Borel, bρ∈Lp(U), and that the upper gradient inequality holds for the triples (u, bρ, γ)for every rectifiable path. Let Γ2denote the collection of all paths γ: [a, b]→Usatisfying ˆ γbρ ds =∞. In particular, if γ∈ Γ2is absolutely continuous, then u◦γis absolutely continuous. Note also that if γcontains a subpath in Γ0∪Γ1, then γ∈Γ2. Moreover, as bρis Lp(U)-integrable, Γ2is p-negligible. In case p=∞,bρ∈L∞(U)and we may apply Theorem 3.6 to deduce that for almost all t∈u(U), ˆ u−1(t)∩Ubρ dH1≤ ∥bρ∥∞H1(u−1(t)∩U)<∞. For 2≤p < ∞, we claim that for almost every t∈u(U), ˆ u−1(t)∩U (bρ)p−1dH1<∞.(20) Indeed, we apply the coarea inequality Theorem 3.6 to the Borel function g= (bρ)p−1and the 1-weak upper gradient bρof u. Then Hölder’s inequality and (20) imply ˆ u−1(t)∩Ubρ dH1<∞(21) for almost all t∈u(U). So the conclusion (21) holds for every 2≤p≤ ∞. We are now in a position to establish Claims (1) to (4). We establish Claim (4) first. To this end, consider an absolutely continuous γ: [a, b]→ u−1(t)∩Ufor an arbitrary tsatisfying the conclusion (21). Then, as bρ≥ ∞ · χB, we conclude H1(B∩u−1(t)) = 0. Therefore ´γχBds = 0 by definition of the path integral. Next, in addition, we assume that #(γ−1(x)) ≤Mfor H1-almost every x∈u−1(t)∩U. Then the definition of the path integral implies ˆ γbρ ds ≤Mˆ u−1(t)∩Ubρ dH1. Thus, if tsatisfies the conclusion (21), then γ∈ Γ2. Then conclusion (4) follows for any γ as above. In particular, Claim (4) holds. COAREA INEQUALITY FOR MONOTONE FUNCTIONS ON METRIC SURFACES 21 Next, since bρ≥ϵ, we conclude H1(u−1(t)∩U)<∞for every tsatisfying the conclusion (21). In particular, Claim (2) holds. Let Gdenote the collection of t∈u(U)satisfying conclusion (21). We assume for now that u−1(t) = u−1(t)∩Ufor every t∈ G. We prove the latter in the next paragraph. We then show how to establish (1). Let F ⊂ G denote the collection of tfor which u−1(t) contains a non-simple continuum. Then Fis countable by Lemma 2.8. Claim (1) follows from this observation. Next, we claim that whenever t∈ G, then u−1(t) = u−1(t)∩U. To this end, we first observe that every connected component of u−1(t)∩Uhas positive H1-measure by Proposition 3.7. We consider an arbitrary connected component Eof u−1(t)∩U. Then Lemma 2.10 implies the existence of continua (En)∞ n=1 such that En⊂En+1 ⊂E,E=S∞ n=1 Enand H1(E1)>0. Then, by Lemma 2.6, there exists a surjective Lipschitz γn: [0,1] →Ensatisfying 1≤#(γ−1 n(x)) ≤2for H1-almost every x∈En. As in the proof of Claim (4), we conclude γn∈ Γ2. This yields that u◦γnis absolutely continuous. As γnhas zero length in B, by considering a constant speed reparametrization of γn, we may therefore assume that γ−1 n(B)has negligible Lebesgue measure. Also, as uis continuous at every γn(s)for every s∈[0,1] \γ−1 n(B), the absolute continuity of u◦γnimplies u◦γn(s) = tfor every s∈[0,1]. In particular, En⊂u−1(t)for every n∈N. The conclusion u−1(t) = u−1(t)∩U follows by the arbitrariness of n∈Nand the component E. To finish, we show Claim (3). Now outside a countable family of tsatisfying conclusion (21), every continuum E⊂u−1(t)∩Uis simple. When diam E > 0, such sets can be parametrized by a Lipschitz path that is injective outside its end points, as we recall from Lemma 2.4. Thus Claim (3) follows. Since Claims (1) to (4) were proved, the proof is complete. □ We are now ready to prove Theorem 1.10. Proof of Theorem 1.10.It suffices to prove continuity for u:U→Rsatisfying the standing assumptions of this section. Let x0∈U, and consider the numbers s1= lim infy→x0u(y) and s2= lim supy→x0u(y). Then x0is a point of discontinuity for uif and only if s1< s2. We assume that uis discontinuous at x0and derive a contradiction. To this end, from Lemma 3.3 we obtain ϵ0>0and a nested collection of Jordan domains Urcompactly contained in U, for almost every 0< r < ϵ0, for which u|∂Uris continuous, d(x0, y) = r for every y∈∂Urand TrUr={x0}. Note that the continuity of u|∂Urfollows from the existence of a Lipschitz parametrization of ∂Urinjective outside its end points, such that uis absolutely continuous along the parametrization. Continuity of u|∂Urimplies that u(∂Ur)is connected. Also, Definition 1.9 implies that u(∂Ur)⊃(s1, s2). Since ris arbitrary, we conclude x0∈u−1(t)for every s1<t<s2. Proposition 3.7 yields the existence of a connected component Et⊂u−1(t)∩Ucontaining x0, with diam Et>0. On the other hand, Proposition 3.9 implies u|Et=tfor almost every such t. This is a contradiction since (s1, s2)has positive measure. □ 22 BEHNAM ESMAYLI, TONI IKONEN, KAI RAJALA Example 3.10. Assumption p≥2in Theorem 1.10 cannot be relaxed even in the standard R2. Indeed, u:D→Rdefined by u(x) = x1+x1/|x|for x= 0 and u(0) = 0 is weakly monotone and discontinuous at the origin. Moreover, u∈D1,p(D)for all p < 2. See [IKO01, Sect. 3] for further details. 4. Coarea inequality via weakly quasiconformal mappings In this section we give a second proof of the coarea inequality for monotone functions using suitable parametrizations of the metric surfaces from Euclidean domains. The motivation arises from the fact that in the Euclidean setting, better than an inequality, we actually have an equality, known as the the coarea formula (Theorem 1.2). 4.1. Weakly quasiconformal maps. Definition 4.1 (Quasiconformal maps).Given metric spaces Ωand X, endowed with locally finite H2-measures, a homeomorphism f: Ω →Xis quasiconformal if there exists aK≥1such that K−1mod Γ ≤mod fΓ≤Kmod Γ, for every family Γof continuous paths. Here fΓdenotes the collection of all f◦γfor which γ∈Γ. We shall say K-quasiconformal to emphasize the role of K. The third named author established in [Raj17] necessary and sufficient conditions for a domain U⊂Xhomeomorphic to R2in a metric surface Xto admit a quasiconformal parametrization from R2or the disk D. That is, there to exist a quasiconformal homeomorphism φ: Ω →Ufor Ω = Dor Ω = R2. As a sufficient condition, we note the following, sup x∈U lim sup r→0+ H2(B(x, r)) πr2<∞,see [RRR21]. Romney observed in [Rom19] that whenever such a parametrization exists, there exists aπ/2-quasiconformal homeomorphism φ:U→Ω⊂R2. More generally, we say that U⊂Xis a quasiconformal surface if every point x0∈Uis contained in an open set U′⊂Xwhich admits a quasiconformal parametrization. It is now understood that every quasiconformal surface is a π/2-quasiconformal image of a Riemannian surface [Iko21]. In fact, for every quasiconformal surface X, there exists a Riemannian surface Yand a quasiconformal homeomorphism f:X→Ysatisfying 2 πmod Γ ≤mod fΓ≤4 πmod Γ (22) for every path family Γ. Both inequalities are best possible. In general, there are geometric obstructions for a metric surface to be a quasiconformal surface. A typical example involves considering a length space Xobtained from the plane R2by collapsing the closed disk Dto a point and endowing the quotient space with the induced length distance. No neighbourhood of the collapsed disk on Xcan be quasiconformal homeomorphic to a subset of the plane. More subtle examples were recently considered in [IR22,Iko22,NR22]. Fortunately, every metric surface is a weakly quasiconformal image of a smooth Riemannian surface. Before formulating the precise statement, we need a definition. COAREA INEQUALITY FOR MONOTONE FUNCTIONS ON METRIC SURFACES 23 Definition 4.2 (Weakly quasiconformal parametrization.).Given metric spaces Vand U, endowed with locally finite H2-measures, a map f:V→Uis weakly K-quasiconformal if it satisfies the following: (a) fis a uniform limit of homeomorphisms g:V→U, (b) there exists a K≥1such that mod Γ ≤Kmod fΓ,(23) for every path family Γ. Recently, Ntalampekos and Romney established the following result which was conjectured by the third named author and Wenger, cf. [IR22, Question 1.1]. Theorem 4.3 (Theorem 1.3, [NR22]).If Uis a metric surface, then there exists a Riemannian surface Vand a weakly (4/π)-quasiconformal f:V→U. Theorem 4.3 was proved earlier by Meier and Wenger [MW21] and Ntalampekos and Romney [NRar], under the assumption that Uis locally geodesic. See also [LW17]. Observe that the mapping fin Theorem 4.3 satisfies only the upper bound in the stronger result (22). Moreover, the example above shows that the one-sided inequality (23) cannot be upgraded to quasiconformality. Properties of weakly quasiconformal mappings have been extensively studied in the metric surface setting in [NRar] and in greater generality in [Wil12,ILP21]. In particular, we have the following. Lemma 4.4 (Theorem 7.1, [NRar]).Let f:V→Ube a continuous map between metric surfaces. Then fsatisfies mod Γ ≤Kmod(fΓ),for all path families, if and only if fhas a locally 2-integrable 2-weak upper gradient ρfor which ˆ f−1(E) ρ2dH2≤KH2(E)for every Borel set E⊂U.(24) Note, in particular, that if (24) holds for some locally 2-integrable 2-weak upper gradient, it also holds for the minimal 2-weak upper gradient ρf. Consider next ν(E):= ´f−1(E)ρ2 fdH2for every Borel set E⊂U. Inequality (24) is equivalent to requiring that ν is locally finite and that ν(E)≤KH2(E)for all Borel E⊂U. This implies that ˆ U g dν ≤Kˆ U g dH2for any Borel function g:U→[0,∞]. In other words, ˆ V g(f(x))ρ2 f(x)dH2(x)≤Kˆ X g dH2for any Borel function g:U→[0,∞]. (25) 24 BEHNAM ESMAYLI, TONI IKONEN, KAI RAJALA 4.2. Pullback of monotone functions by weakly quasiconformal maps. In this section, f:V→Uis weakly K-quasiconformal where V⊂R2is open and simply connected and Uis a metric surface satisfying H2(U)<∞. Later, u:U→Rwill be a (weakly) monotone function in the sense of Definition 1.9. Throughout, we stick to the notation v:=u◦f. The aim of this section is to prove that vinherits important regularity and structural properties of u. The modulus inequality (23) allows us to pullback Dirichlet functions using f. More precisely, we have the following. Lemma 4.5. Let Vand Ube metric surfaces and f:V→Uweakly quasiconformal. If u:U→[−∞,∞]has a 2-integrable 2-weak upper gradient ρ, then ρ′(x) = ρ(f(x))ρf(x)is a2-integrable 2-weak upper gradient of v:=u◦f. In particular, ρv(x)≤ρu(f(x))ρf(x)for H2-a.e. x∈V. Proof. Let Γ0denote the family of paths γ: [a, b]→Ufor which either (1) u◦γfails to be absolutely continuous, (2) ´γρ ds =∞, or (3) there exists an interval I⊂[a, b]such that ℓ(u◦γ|I)>´I(ρ(γ(t)))|γ′|(t)dt. Then Γ0has negligible 2-modulus since ρis a 2-integrable 2-weak upper gradient of u. In particular, for every absolutely continuous Γ0∋ γ: [a, b]→U, |(u◦γ)′|(t)≤ρ(γ(t))|γ′|(t)for almost every t∈[a, b]; see, e.g., [HKST15, Proposition 6.3.3]. Let Γ1denote the family of paths γ: [a, b]→Vfor which (1) f◦γ∈Γ0, (2) f◦γfails to be absolutely continuous, (3) ´γρfds =∞, or (4) there exists an interval I⊂[a, b]with ℓ(f◦γ|I)>´I(ρf(γ(t)))|γ′|(t)dt. Then Γ1has negligible 2-modulus since fis weakly quasiconformal. Now, for each absolutely continuous Γ1∋ γ: [a, b]→V, we have |(v◦γ)′|(t)≤ρ(f(γ(t)))|(f◦γ)′|(t)≤ρ(f(γ(t)))ρf(γ(t))|γ′|(t)for a.e. t∈[a, b], where [HKST15, Proposition 6.3.3] was used again. Hence ℓ(v◦γ)≤ˆ γ (ρ◦f)ρfdt =ˆ γ ρ′dt. So, ρ′(x) = ρ(f(x))ρf(x)is a 2-integrable 2-weak upper gradient — the integrability follows from (25). □ Lemma 4.6. Suppose that Vand Uare metric surfaces and f:V→Uis a uniform limit of homeomorphisms fn:V→U. If u:U→Ris a monotone function, then v=u◦fis a monotone function. COAREA INEQUALITY FOR MONOTONE FUNCTIONS ON METRIC SURFACES 31 minimal upper gradient of any function is zero (a.e.) on C. Therefore, if ρis the minimal upper gradient of some u:X→R, then ˆ C ρ dH2= 0 .(36) Now, as Cis a four-corner Cantor set with H2(C)>0, and u(x1, x2, x3) = x1is the orthogonal projection, by Fubini’s theorem ˆ R H1(u−1(t)∩C)dt > 0. By (36), thus, the coarea inequality fails for the Lipschitz uand any minimal upper gradient of it. The equality (35) follows for Borel sets E⊂X\Cfrom the smoothness of X\C. On the other hand, as C⊂R2× {0}, we may consider Borel subsets E⊂Cas a subset of the plane. The planar coarea formula then yields ˆ R H1(u−1(t)∩E)dt =ˆ E lip (u|C)dH2for every Borel E⊂C; the equality follows by applying the planar coarea formula for a Lipschitz extension of u|C. Then (35) follows from the smoothness of X\C.□ Remark 5.4. The last paragraph of the proof of Theorem 5.3 could also be argued using the coarea formula established by Ambrosio and Kirchheim, cf. [AK00, Theorem 9.4]. Acknowledgement. We thank Sylvester Eriksson–Bique for pointing out an initial inaccuracy in the construction of Section 5. We thank the referee for very helpful comments. References [ACDM15] Luigi Ambrosio, Maria Colombo, and Simone Di Marino. Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope. In Variational methods for evolving objects, volume 67 of Adv. Stud. Pure Math., pages 1–58. Math. Soc. Japan, [Tokyo], 2015. [AGS13] Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. 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Reprint of 1963 edition. [Wil12] Marshall Williams. Geometric and analytic quasiconformality in metric measure spaces. Proc. Amer. Math. Soc., 140(4):1251–1266, 2012. B. Esmayli: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, University of Jyväskylä, Finland. [email protected] T. Ikonen: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, University of Jyväskylä, Finland. [email protected] K. Rajala: Department of Mathematics and Statistics, University of Jyväskylä, P.O. Box 35 (MaD), FI-40014, University of Jyväskylä, Finland. [email protected]