On the Modulus Duality in Arbitrary Codimension

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 4.0 https://creativecommons.org/licenses/by/4.0/ On the Modulus Duality in Arbitrary Codimension © The Author(s) 2022. Published by Oxford University Press Published version Lohvansuu, Atte Lohvansuu, A. (2022). On the Modulus Duality in Arbitrary Codimension. International Mathematics Research Notices, Advance article. https://doi.org/10.1093/imrn/rnac238 2022 A. Lohvansuu (2022) “On the Modulus Duality in Arbitrary Codimension,” International Mathematics Research Notices, Vol. 00, No. 0, pp. 1–18 https://doi.org/10.1093/imrn/rnac238 On the Modulus Duality in Arbitrary Codimension Atte Lohvansuu∗ Department of Mathematics and Statistics, University of Jyväskylä, PL 35, 40014 Jyväskylän yliopisto, Finland ∗Correspondence to be sent to: e-mail: atte.s.lohv[email protected]i We study the modulus of dual families of k-and(n−k)-dimensional Lipschitz chains of Euclidean n-cubes and establish half of the modulus duality identity. 1 Introduction Suppose D⊂R2is a Jordan domain, whose boundary is divided into four segments ζ1,...,ζ4, in cyclic order. Let (ζ1,ζ3;D)be the family of all paths of Dthat connect ζ1 and ζ3. Then for every 1 <p<∞ (modp(ζ1,ζ3;D))1/p(modq(ζ2,ζ4;D))1/q=1. (1) Here q=p p−1and the p-modulus of a path family is defined by modp=inf ρD ρpdH2, where the infimum is taken over all positive Borel-functions ρwith γ ρds≥1 Communicated by Prof. Assaf Naor Received June 1, 2021; Revised July 8, 2021; Accepted August 7, 2021 © The Author(s) 2022. Published by Oxford University Press. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited. 2 A. Lohvansuu for every locally rectifiable path γ∈. The path modulus is a fundamental tool in geometric function theory and nonsmooth analysis [10,18,22]. To prevent confusion, we use the plural form of the word modulus, moduli, sparingly. We emphasize that the concepts and results of this paper are not closely related to moduli spaces. For conformal moduli, that is, p=2=q, the duality relation (1) was already known to Beurling and Ahlfors, see for example, [1, Lemma 4] and [2, Ch. 14], although instead of modulus they considered its reciprocal, called extremal length. For general p the identity (1) follows from the results of [24]. It has found applications in connection with uniformization theorems [12,17] and Sobolev extension domains [23]. The duality property of the modulus is also present in Euclidean spaces of higher dimension [6,8,24] and sufficiently regular metric spaces [13–15]. Moreover, discrete analogues can be found in the context of graphs and networks, see [3]andthe references therein. For example, in [24], it is shown that for 1 <p<∞ (modp(E,F;G))1/p(modq∗(E,F;G))1/q=1, (2) where G⊂Rnis open and connected, Eand Fare disjoint, compact and connected subsets of Gand ∗(E,F;G)is the set of all relatively closed sets of Gthat separate E from F. The modulus of separating sets is a natural generalization of the path modulus. See Section 2for definitions of moduli and other concepts appearing in the introduction. Separating sets are generally of codimension 1, so (1)and(2) deal with objects of either dimension or codimension 1. In fact, this is a common theme in all of the results cited above. Indeed, not much is known about the relationship of the modulus and objects of higher (co)dimension. However, the modulus has been applied to study such objects in [11,16], where the nonexistence of quasisymmetric parametrizations of certain spaces was established. Studying more general moduli could therefore lead to finding tools to approach such parametrization questions in higher dimensions. An observation by Freedman and He (see the discussion after Theorem 2.5 in [6]) hints that a duality result could hold for objects of higher (co)dimension as well. In this paper we explore this question in the setting of cubes of Rn. Our first problem is defining suitable classes of k-and(n−k)-dimensional objects, since simple descriptions such as “connecting paths” or “separating surfaces” do not seem to exist. We follow [6] and define the objects as representatives of certain relative homology classes. For example, in the context of (1), we can think of the paths of (ζ1,ζ3;D)as singular relative cycles, which are representatives of either generator of H1(D,ζ1∪ζ3)≃Z. Since we also want to integrate over the chains, we need to assume On the Modulus Duality in Arbitrary Codimension 3 some regularity. For this reason we will consider Lipschitz chains instead of singular chains. Let Q⊂Rnbe a compact set homeomorphic to the closed unit n-cube In.Fixa homeomorphism h:Q→Inand an integer 0 <k<n,andlet A=h−1(∂Ik×In−k)and B=h−1(Ik×∂In−k). Then Aand Bare (n−1)-dimensional submanifolds of ∂Qwith ∂Q=A∪Band ∂A=A∩B=∂B. We assume that A,B,andQare locally Lipschitz neighborhood retracts. This includes triples (Q,A,B)that are smooth or polygonal and cubes that are images of the standard cube under bi-Lipschitz automorphisms of Rn. We denote the Lipschitz homology groups by HL ∗. We consider only groups with integer coefficients. This notation should not be confused with the Hausdorff measures, which are denoted by H∗.Notethat HL k(Q,A)≃Z≃HL n−k(Q,B), since the same is true for singular homology, and the two homology theories are equivalent for pairs of locally Lipschitz retracts (see Lemma 2.1). Let A(resp. B) be the collection of the images of relative Lipschitz k-cycles of Q−Bthat generate HL k(Q,A)((n−k)-cycles of Q−Athat generate HL n−k(Q,B)). Define modpA:=inf ρQ ρpdHn, where the infimum is taken over positive Borel-functions ρ,forwhich S ρdHk≥1 for every S∈A. The modulus modpBis defined analogously. In this paper we will prove the following upper bound. Theorem 1.1. For every 1 <p<∞ (modpA)1/p(modqB)1/q≤1, where q=p p−1. 4 A. Lohvansuu It is unknown whether Theorem 1.1 holds with an equality. We will prove Theorem 1.1 in Section 3. A similar result for de Rham cohomology classes, with an equality, is proved in the setting of Riemannian manifolds in pages 212–213 of [6]. In light of the results of [5, Ch. 4], it would be interesting to know whether analogues of Theorem 1.1 hold for homology classes of integral currents. Remark 1.2. The assumption on Q,A,andBbeing locally Lipschitz neighborhood retracts can be relaxed. The proof of Theorem 1.1 only requires that there exists a pair of Lipschitz chains that generate Hk(Q,A)and Hn−k(Q,B). The assumption on retracts was chosen for its simplicity and its use in [5]. It is also likely that such minimal assumptions for the upper bound of Theorem 1.1 are not sufficient for the corresponding lower bound. We will discuss the lower bound in Section 4. 2 Definitions 2.1 Lipschitz homology Let us recall the definition and basic properties of the integral homology groups. See for example [4,9] or other texts on basic algebraic topology for a more comprehensive treatment. For an integer k≥0 the standard k-simplex kis the convex hull of the standard unit vectors e0,...,ekof Rk+1. Given a metric space (X,d), a singular k-simplex is a continuous map from kto X. Finite formal linear combinations σ= i kiσi of singular k-simplices σiwith integer coefficients kiare called singular k-chains. Singular k-chains of Xform a free abelian group denoted by Ck(X). The boundary ∂σ of a singular k-simplex σis the singular (k−1)-chain ∂σ = k  i=0 (−1)iσ◦Fi k, where Fi k:k−1→kis the unique linear map that maps each ejto ejfor j<iand to ej+1for j≥i. For singular 0-simplices we set ∂σ =0. The boundary defines a collection of homomorphisms ∂:Ck(X)→Ck−1(X), all denoted by the same symbol ∂.Then∂∂ =0. The image of a singular k-simplex σis the compact set |σ|=σ(k). The image of a k-chain σ=ikiσiis the compact set |σ|=i|σi|. On the Modulus Duality in Arbitrary Codimension 5 Given a subspace Y⊂X, we identify each singular simplex σof Ywith the singular simplex iY◦σof X, where iY:Y→Xis the inclusion map. We define the groups of relative chains by Ck(X,Y):=Ck(X) Ck(Y), with the convention Ck(X,∅)=Ck(X). The boundary map induces homomorphisms ∂:Ck(X,Y)→Ck−1(X,Y), which are again denoted by the same symbol. A chain σ∈Ck(X)is called a cycle relative to Y,if∂σ ∈Ck−1(Y), or simply a relative cycle if the choice of Yis clear from the context. Similarly, σis called a relative boundary if σ=∂σ+σ, where σ∈Ck+1(X)and σ ∈Ck(Y). The singular relative homology groups of the pair (X,Y)are the quotient groups Hk(X,Y):=ker(∂ :Ck(X,Y)→Ck−1(X,Y)) im(∂ :Ck+1(X,Y)→Ck(X,Y)) . The homology groups of Xare the groups Hk(X):=Hk(X,∅). The homology class of a (relative) chain σis denoted by [σ]. The homology classes of Hk(X,Y)are represented by relative k-cycles, and two relative k-cycles define the same class if and only if their difference is a relative boundary. If Xis another metric space with a subset Y,andf:X→Xis a continuous map with f(Y)⊂Y, we denote by f∗the induced homomorphisms f∗:Ck(X,Y)→Ck(X,Y), and also the homomorphisms f∗:Hk(X,Y)→Hk(X,Y). These are given by f∗σ=f◦σ for singular simplices, f∗ikiσi=ikif∗σifor chains and f∗[σ]=[f∗σ]forhomology classes. Given a continuous homotopy H:X×I→Xwith H(Y×I)⊂Y, there exists a sequence of homomorphisms P:Ck(X,Y)→Ck+1(X,Y), such that H1∗−H0∗=P∂+∂P.(3) Here Ht(x)=H(x,t). Formula (3) is called the homotopy formula. A continuous f:X→Yis called a retraction if f◦iY=idY. The set Yis then called a retract of X.IfYis a retract of one if its neighborhoods in X, it is called a neighborhood retract. 6 A. Lohvansuu The corresponding objects in the Lipschitz category are obtained by replacing each occurrence of “singular” or “continuous” with “Lipschitz.” The homotopies involved in these definitions are then required to be Lipschitz with respect to the metric d((x,t),(x,t)) =d(x,x)+|t−t|. We denote the groups of Lipschitz chains by CL ∗(X,Y) and the Lipschitz homology groups by HL ∗(X,Y). We define locally Lipshitz objects similarly. However, due to compactness, there is often no difference between the corresponding objects of Lipschitz and locally Lipschitz categories. Lemma 2.1. Let Y⊂X⊂Rnbe locally Lipschitz neighborhood retracts of Rn.Then the inclusions i:CL ∗(X,Y)→C∗(X,Y) induce isomorphisms on homology. Lemma 2.1 follows from a more general result [19,Cor. 11.1.2], which holds for pairs of locally Lipschitz contractible metric spaces (see [19] for the definition). It is straightforward to show that the existence of locally Lipschitz neighborhood retractions implies locally Lipschitz contractibility. 2.2 Modulus Given a 1 <p<∞and a family Mof Borel measures of Rn,thep-modulus of Mis the number modpM:=inf ρRn ρpdHn,(4) where the infimum is taken over all Borel functions ρ:Rn→[0, ∞]with Rn ρdν≥1(5) for every ν∈M. Such functions are called admissible for M. If there exists a subfamily N⊂Msuch that modpN=0and(5)holdsforallν∈M−N,wesaythatρis p-weakly admissible or simply weakly admissible if the choice of pis clear from the context. It follows that the infimum in (4) does not change if we take it over p-weakly admissible functions instead. Let us list some useful properties of the modulus. On the Modulus Duality in Arbitrary Codimension 7 Lemma 2.2. Let Mbe a collection of Borel measures of Rn.Let1<p<∞. i) If ρiare p-integrable Borel functions that converge to a function ρin Lp, there exists a subsequence (ρij)jfor which Rn ρijdνj→∞ −→ Rn ρdν for almost every ν∈M. In particular, Borel representatives of Lp-limits of admissible functions are weakly admissible. ii) If modpM<∞,then modpM=Rn ρpdHn for a weakly admissible minimizer ρ, unique up to sets of Hn-measure zero. Moreover, modpM≤Rn φρp−1dHn for any other p-integrable weakly admissible φ. iii) If M=∞ i=1Miwith Mi⊂Mi+1for all i,then modpM=lim i→∞ modpMi. Claim i)is often referred to as Fuglede’s lemma. Proofs for i)and the first part of ii)can be found in [7, Thm. 3]. The second part of ii)and iii)are generalizations of [15, Lemma 5.2] and [25, Lemma 2.3], respectively. The same proofs apply. In this paper we abbreviate modpA=modp{HkS|S∈A}, and modqB=modq{Hn−kS∗|S∗∈B}. 2.3 Rectifiable sets A subset of Rnis k-rectifiable if it is covered by the image of a subset of Rkunder a Lipschitz map. A subset of Rnis countably k-rectifiable if Hk-almost all of it is contained in a countable union of k-rectifiable sets. 8 A. Lohvansuu See for example [5,21] for basic theory on rectifiable sets. Note that the definition of countable rectifiability in [5, 3.2.14] is slightly different from ours. Let us record some useful facts on rectifiable sets. The following Fubini-type lemma is an application of [5, 3.2.23] and [5, 2.6.2]. Lemma 2.3. Suppose S∗is a countably k-rectifiable subset of Rnand Sis a countable union of l-rectifiable subsets of Rm.ThenS∗×Sis a countably (k+l)-rectifiable subset of Rn×Rm,and S∗×S g(x,y)dHk+l(x,y)=S∗S g(x,y)dHl(y)dHk(x) for any positive Borel function gon Rn×Rm. Lemma 2.3 does not hold for general countably l-rectifiable sets S, see [5, 3.2.24]. The second tool we need is the coarea formula, see for example, [21,12.7]. Lemma 2.4. Suppose m≤k.LetSbe a countably k-rectifiable subset of Rnand let u:Rn→Rmbe locally Lipschitz. Then Rmu−1(z)∩S gdHk−mdHm(z)=S gJS udHk(6) for every positive Borel function gon S. Let us define the Jacobian JS uappearing in (6). Details can be found in [21, §12]. Suppose first that Sis an embedded C1k-submanifold (without boundary) of Rn.Then uis differentiable at Hk-almost every x∈S. Fix such an x,andlet{E1,...,Ek}be an orthonormal basis for the tangent space of Sat x.LetDu(x)be the Jacobian matrix of u at xwith respect to standard bases of Rnand Rm. We set JS u(x):=det(dSu(x)dSu(x)t), where dSu(x)is the matrix with columns Du(x)Ei. It can be shown that JS u(x)does not depend on the choice of the basis {Ei}. More generally, every countably k-rectifiable set Scan be expressed as a disjoint union S=∞ i=0Mi, where Hk(M0)=0 and each Mifor i≥1 is contained in an embedded On the Modulus Duality in Arbitrary Codimension 15 and the infimum is taken over Lipschitz functions u:Q→Iwith u|A0=0andu|A1=1. Then by the coarea formula 1≤Iu−1(t) ρdHn−1dt=Q ρ|∇u|dHn for any integrable ρadmissible for ∗ A,sinceby[5, 3.2.15] almost every level set u−1(t) is an element of ∗ A. Now the lower bound (14) follows from Hölder’s inequality and (15). Similar ideas can be used to prove that Theorems 1.1 and 3.1 are sharp for any n and k. Let us show that (13) holds whenever Q=Q1×Q2, where Q1⊂Rkand Q2⊂Rn−k are k-and(n−k)-dimensional topological cubes as in Theorem 1.1,A=∂Q1×Q2and B=Q1×∂Q2. Then it suffices to show that modpA=Hn−k(Q2) Hk(Q1)p−1and modqB=Hk(Q1) Hn−k(Q2)q−1. The proofs of the two formulas are identical, so we only consider A. For every y∈Q2 and ρadmissible for A 1≤Q1×{y} ρdHk, so by Hölder’s inequality 1≤Q1×{y} ρpdHk1/p Hk(Q1)1/q, from which we obtain the inequality “≥” by integrating over yand applying Fubini’s theorem (or the coarea formula applied on the projection π2(x,y)=y). The reverse inequality follows from the observation that Hk(Q1)−1χQis admissible for A. It is also noteworthy that in this case modqB=modq∗ A, and both are equal to the q-modulus of the slices {x}×Q2. Observe that if we let λ=Hk(Q1)−1/kand use a scaled projection map λπ1(x,y)=λxinstead, we find that Hk(λπ1(Q1×Q2)) =1andJλπ1=Hk(Q1)−1χQ. That is, the minimizer of modpAis the Jacobian of λπ1. Moreover, the level sets of λπ1 are elements of B. 16 A. Lohvansuu Inspired by this example we extend the definition of the capacity to general Q and Aby cappA:=inf uQ Jp udHn, where the infimum is taken over all such Lipschitz maps u:(Q,A)→(¯ U,∂U),thatUis a domain in Rknormalized with Hk(U)=1, (¯ U,∂U)is homeomorphic to (¯ Bk,∂Bk),and the induced homomorphism u∗:Hk(Q,A)→Hk(¯ U,∂U)≃Z(16) is an isomorphism. We observe that U⊂u(S)for any S∈A, so almost every level set of uis in ∗ A,sinceHk(¯ U−{x},∂U)is trivial for all x∈U. Moreover, the Cauchy–Binet formulaimpliesthatJu≥JS u,so S JudHk≥S JS udHk≥U dHk=1 by Lemma 2.4. Thus Juis admissible for Aand modpA≤cappA. It is unknown whether the reverse inequality is true, but it would imply (14). To prove the reverse inequality one would have to be able to construct the required Lipschitz maps u. This seems to be very difficult when k>1, especially with a given Ju.Ifk=1, the situation is considerably simpler, since then Ju=|∇u|and the unit interval Iis practically the only choice of U. 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