Classification criteria for regular trees

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 4.0 https://creativecommons.org/licenses/by-nc/4.0/ Classification criteria for regular trees © 2021 Annales Fennici Mathematici Published version Nguyen, Khanh Nguyen, K. (2022). Classification criteria for regular trees. Annales Fennici Mathematici, 47(1), 3- 21. https://doi.org/10.54330/afm.112449 2022 Annales Fennici Mathematici Volumen 47, 2022, 3–21 Classification criteria for regular trees Khanh Nguyen Abstract. We give characterizations for the parabolicity of regular trees. Luokitusehtoja säännöllisille puille Tiivistelmä. Esitämme säännöllisten puiden parabolisuudelle yhtäpitäviä ehtoja. 1. Introduction Let us begin with the uniformization theorem of Klein, Koebe and Poincaré for Riemann surfaces. The celebrated theorem says that every simply connected Riemann surface Mis conformally equivalent (or bi-holomorphic) to one of three Riemann surfaces: the half plane H2(surface of hyperbolic type), the complex plane R2(surface of parabolic type), the Riemann sphere S(surface of elliptic type). Then Madmits a Riemannian metric gwith constant curvature. A simply connected Riemann surface is said to be hyperbolic if it is conformally equivalent to H2, otherwise we say that it is parabolic. Let Mbe a simply connected Riemann surface with Riemannian metric g. A C2-smooth function udefined in Mis superharmonic if −∆u≥0 where ∆is the Laplace–Beltrami operator associated to the Riemannian metric g. It is well known that every conformal mapping in dimension two preserves superharmonic functions (see [1, Page 135]). Since H2possesses a nonconstant nonnegative superharmonic function and every nonnegative superharmonic function on R2or S is constant, it then follows that there is no nonconstant nonnegative superharmonic function on (M, g)if and only if Mis parabolic. Let Kbe a compact subset in (M, g). We define the capacity Cap(K)by Cap(K) = inf ˆM |∇u|2dmg:u∈Lip0(M), u|K≡1 where Lip0(M)is a set of all Lipschitz functions with compact support on M, and mg is the Riemannian measure associated to g. Then there is a nonconstant nonnegative superharmonic function on Mif and only if Cap(K)>0for some compact subset K, (see [7, Theorem 5.1] for Riemannian manifolds). It follows that the parabolicity of a Riemann surface Mcan be characterized both in terms of capacity and superharmonic functions. By this reason, in the setting of Riemannian manifolds or metric measure spaces, one defines parabolicity either via capacity (see [12, 14, 15, 16]) or via superharmonic functions (see [7] and also references therein). In this paper, https://doi.org/10.54330/afm.112449 2020 Mathematics Subject Classification: Primary 31C05, 31C15, 31C45, 31E05. Key words: Capacity, harmonic function, parabolicity, regular tree. The author has been supported by the Academy of Finland (project No. 323960). c 2022 The Finnish Mathematical Society 4Khanh Nguyen we will consider K-regular trees and give the definition of parabolicity in terms of capacity. Recently, analysis on K-regular trees has been under development, see [3, 21, 22, 23, 20, 27]. Let Gbe a K-regular tree with a set of vertices Vand a set of edges Efor some K≥1. The union of Vand Ewill be denoted by X. We abuse the notation and call XaK-regular tree. We introduce a metric structure on X by considering each edge of Xto be an isometric copy of the unit interval. Then the distance between two vertices is the number of edges needed to connect them and there is a unique geodesic that minimizes this number. Let us denote the root by 0. If xis a vertex, we define |x|to be the distance between 0and x. Since each edge is an isometric copy of the unit interval, we may extend this distance naturally to any xbelonging to an edge. We define ∂X as the collection of all infinite geodesics starting at the root 0. Then every ξ∈∂X corresponds to an infinite geodesic [0, ξ)(in X) that is an isometric copy of the interval [0,∞). Let µand λ: [0,∞)→(0,∞)be locally integrable functions. Let d|x|be the length element on X. We define a measure µon Xby setting dµ(x) = µ(|x|)d|x|, and a metric don X via ds(x) = λ(|x|)d|x|by setting d(y, z) = ´[y,z]ds whenever y, z ∈Xand [y, z]is the unique geodesic between yand z. Then (X, d, µ)is a metric measure space and hence one may define a Newtonian Sobolev space N1,p(X) := N1,p(X, d, µ)based on upper gradients [10, 24]. As usual, N1,p 0(X)is the completion of the family of functions with compact support in N1,p(X), and ˙ N1,p 0(X)is the completion of the family of functions with compact support in ˙ N1,p(X), the homogeneous version of N1,p(X). Let Ωbe a subset of X. We denote by N1,p loc (Ω) the space of all functions u∈Lp loc(Ω) that have an upper gradient in Lp loc(Ω), where Lp loc(Ω) is the space of all measurable functions that are p-integrable on any compact subset of Ω. See Section 2 for the precise definitions. Let 1< p < ∞and Obe a subset of X. We define the p-capacity of O, denoted Capp(O), by setting (1.1) Capp(O) = inf ˆX gp udµ:u|O≡1, u ∈N1,p 0(X) where guis the minimal upper gradient of uas in Section 2.2. A K-regular tree X is said to be p-parabolic if Capp(O) = 0 for all compact sets O⊂X; otherwise Xis p-hyperbolic. Given 1< p < ∞and an open subset Ω⊆X, we say that u∈N1,p loc (Ω) is a p-harmonic function (or a p-superharmonic function) on Ωif (1.2) ˆspt(ϕ) gp udµ ≤ˆspt(ϕ) gp u+ϕdµ holds for all functions (or for all nonnegative functions) ϕ∈N1,p(Ω) with compact support spt(ϕ)⊂Ω. We refer the interested readers to [2, 9, 11] for a discussion on the p-capacity and p-(super)harmonic functions. Since a K-regular tree (X, d)is the quintessential Gromov hyperbolic space, it is then natural to ask for whether the parabolicity (or hyperbolicity) of Xcan be characterized via p-(super)harmonic functions under some conditions on the measure µonly depending on the given metric d, and also ask for intrinsic conditions of K- regular trees that would characterize the parabolicity (or hyperbolicity). We refer the readers to [1, Chapter IV] for a discussion in the case of Riemann surfaces, and Classification criteria for regular trees 5 [6, 7, 14, 15, 16] for a discussion in the setting of Riemannian manifolds, and [25, Section 6], [28] for a discussion on infinite networks. In order to state our results, we introduce a notion from [21]. Let 1< p < ∞. We set Rp(λ, µ) = ˆ∞ 0 λ(t)p p−1µ(t)1 1−pKj(t) 1−pdt where j(t)is the smallest integer such that j(t)≥t, and let Xn={x∈X:|x| ≤ n} for each n∈N. Since we work with a fixed pair λ, µ, we will usually write Rp(λ, µ) simply as Rpwhen no confusion can arise. In what follows, we additionally assume that λpµ−1∈L1/(p−1) loc ([0,∞)) to make sure that the finiteness of Rpis a condition at infinity. The first result of our paper is a characterization of parabolicity of K-regular trees. Theorem 1.1. Let 1< p < ∞and Xbe a K-regular tree with metric dand measure µas above, with K≥1. Then (X, d, µ)is p-parabolic if and only if any one of the following conditions is fulfilled: 1. Rp(λ, µ) = ∞. 2. Capp(Xn) = 0 for all n∈N∪ {0}. 3. Capp(Xn) = 0 for some n∈N∪ {0}. In Section 2.1, we will show that the compactness on a K-regular tree Xwith respect to our metric dand with respect to the graph metric are equivalent. Since each compact set in (X, d)is contained in some n-level set Xnthat is an analog of a ball with respect to graph metric, parabolicity of Xcan be characterized by the zero p-capacity of some/ all n-level sets Xn. In [21, Theorem 1.3], the condition Rp(λ, µ) = ∞gives a characterization of the existence of boundary trace operators and for density properties for ˙ N1,p(X). Hence parabolicity of K-regular trees can be characterized in terms of boundary trace operators and density properties. Combining Theorem 1.1 and [21, Theorem 1.3 and Theorem 3.5], we obtain the following corollary. Corollary 1.2. Let 1< p < ∞and Xbe a K-regular tree with metric dand measure µas above, with K≥1. Then (X, d, µ)is p-parabolic if and only if any one of the following conditions is fulfilled: 1. There exists u∈˙ N1,p(X)such that lim [0,ξ)∋x→ξu(x) = ∞ for all ξ∈∂X. 2. ˙ N1,p 0(X) = ˙ N1,p(X). It is well known, see for instance the survey paper [15], that the volume growth condition ˆ∞ 1t V(B(0, t))1 p−1 dt =∞ is a sufficient condition to guarantee parabolicity of Riemannian manifolds. Here V(B(0, t)) is the volume of the ball with radius tand center at a fixed point 0. However, this condition is far from being necessary in general, as shown by a counterexample due to Holopainen [14] and to Varopoulos [26] in the case p= 2. Our condition Rp(λ, µ) = ∞is an analog of this volume growth condition. Example 3.8 in 6Khanh Nguyen Section 3 shows that there exists a K-regular tree with a distance dand a “non-radial” measure µsuch that Rp(λ, µ) = ∞but Xis p-hyperbolic. Let 1< p < ∞and let int(Xn) := {x∈X:|x|< n}for n∈N. We say that (int(Xn), d, µ)is doubling and supports ap-Poincaré inequality if there exist constants C1≥1, C2>0only depending on nsuch that for all balls B(x, 2r)⊂ int(Xn), µ(B(x, 2r)) ≤C1µ(B(x, r)) and for all balls B(x, r)⊂int(Xn), − ˆB(x,r) |u−uB(x,r)|dµ ≤C2r− ˆB(x,r) gpdµ1 p whenever uis a measurable function on B(x, r)and gis an upper gradient of u, where uB(x,r):= − ´B(x,r)udµ =1 µ(B(x,r)) ´B(x,r)udµ. The validity of a p-Poincaré inequality for Xhas very recently been characterized via a Muckenhoupt-type condition under a doubling condition on (X, d, µ), see [23] for more information. Our second result deals with a characterization of parabolicity in terms of p- (super)- harmonic functions. Theorem 1.3. Let 1< p < ∞and Xbe a K-regular tree with metric dand measure µas above, with K≥2. Assume additionally that (int(Xn), d, µ)is doubling and supports a p-Poincaré inequality for each n∈N. Then (X, d, µ)is p-parabolic if and only if any one of the following conditions is fulfilled: 1. Every nonnegative p-superharmonic function uon Xis constant. 2. Every nonnegative p-harmonic function uon Xis constant. 3. Every bounded p-harmonic function uon Xis constant. 4. Every bounded p-harmonic function uon Xwith ´Xgp udµ < ∞is constant. Let us close the introduction with some comments on Theorem 1.3. According to a version of Theorem 1.3 in the setting of Riemannian manifolds from [12, 13, 17] we have that 1.⇒2.⇒3.⇒4. However 3.does not imply 2.in general. Our condition that (int(Xn), d, µ)is doubling and supports a p-Poincaré inequality for each n∈N is equivalent to µbeing a locally doubling measure supporting a local p-Poincaré inequality on (X, d). Theorem 1.3 is not empty in the sense that there exist both p-parabolic and p-hyperbolic K-regular trees that are doubling and support a p-Poincaré inequality, see Example 3.9 in Section 3 for more details. The motivation for our paper comes from classification problems of spaces. By the survey papers [4, 7], the development of potential theory in the setting of metric measure spaces leads to a classification of spaces as either p-parabolic or not. This dichotomy can be seen as a non-linear analog of the recurrence or transience dichotomy in the theory of Brownian motion. This classification is helpful in the development of a quasiconformal uniformization theory, or for a deeper understanding of the links between the geometry of hyperbolic spaces and the analysis on their boundaries at infinity. The paper is organized as follows. In Section 2, we introduce K-regular trees, Newtonian spaces, and p-(super)harmonic functions on our trees. In Section 3, we give the proofs of Theorem 1.1 and Theorem 1.3. Throughout this paper, the letter C(sometimes with a subscript) will denote positive constants that usually depend only on the space and may change at different Classification criteria for regular trees 7 occurrences; if Cdepends on a, b, . . ., we write C=C(a, b, . . .). For any function f∈L1 loc(X)and any measurable subset A⊂X, let − ´Afdµ stand for 1 µ(A)´Afdµ. 2. Preliminaries 2.1. Regular trees. Agraph Gis a pair (V, E), where Vis a set of vertices and Eis a set of edges. We call a pair of vertices x, y ∈Vneighbors if xis connected to yby an edge. The degree of a vertex is the number of its neighbors. The graph structure gives rise to a natural connectivity structure. A tree Gis a connected graph without cycles. A graph (or tree) is made into a metric graph by considering each edge as a geodesic of length one. We call a tree Garooted tree if it has a distinguished vertex called the root, which we will denote by 0. The neighbors of a vertex x∈Vare of two types: the neighbors that are closer to the root are called parents of xand all other neighbors are called children of x. Each vertex has a unique parent, except for the root itself that has none. We say that a tree is K-regular if it is a rooted tree such that each vertex has exactly Kchildren for some integer K≥1. Then all vertices except the root of a K-regular tree have degree K+ 1, and the root has degree K. Let Gbe a K-regular tree with a set of vertices Vand a set of edges Efor some integer K≥1. For simplicity of notation, we let X=V∪Eand call it a K-regular tree. For x∈X, let |x|be the distance from the root 0to x, that is, the length of the geodesic from 0to x, where the length of every edge is 1and we consider each edge to be an isometric copy of the unit interval. The geodesic connecting two points x, y ∈X is denoted by [x, y]. Throughout this paper, we denote Xn:= {x∈X:|x| ≤ n}and int(Xn) := {x∈X:|x|< n}for each n∈N. On our K-regular tree X, we define a measure µand a metric dvia ds by setting dµ(x) = µ(|x|)d|x|, ds(x) = λ(|x|)d|x|, where λ, µ: [0,∞)→(0,∞)are fixed with λ, µ ∈L1 loc([0,∞)). Here d|x|is the measure which gives each edge Lebesgue measure 1, as we consider each edge to be an isometric copy of the unit interval and the vertices are the end points of this interval. Hence for any two points z, y ∈X, the distance between them is d(z, y) = ˆ[z,y] ds(x) = ˆ[z,y] λ(|x|)d|x| where [z, y]is the unique geodesic from zto yin X. We abuse the notation and let µ(x)and λ(x)denote µ(|x|)and λ(|x|), respectively, for any x∈X, if there is no danger of confusion. We denote by dEthe graph metric on X. Then for any two points z, y ∈X, dE(z, y) = ˆ[z,y] d|x| is the graph distance between zand ywhere [z, y]is the unique geodesic from zto y. Theorem 2.1. The identity mapping Id X: (X, dE)→(X, d)is a homeomorphism. Proof. Let us first prove that the identity mapping f: (X, dE)→(X, d),f(x) = x if x∈X, is continuous. Let Bd(x, r)be an arbitrary open ball with center xand radius r > 0in (X, d). Recall that λ: [0,∞)→(0,∞)is a locally integrable function. 8Khanh Nguyen Hence λis an integrable function on [a, b]whenever [a, b]is a compact interval with |x| ∈ (a, b)if x6= 0, or |x|=aif x= 0 where 0is the root of X. Then F(h) := ˆh a λ(t)dt is absolutely continuous on [a, b]. It follows that there exists δr>0only depending on x, r such that (´|x|+δr |x|−δrλ(t)dt < r 2if |x| ∈ (a, b), x 6= 0, ´|x|+δr |x|λ(t)dt < r 2if |x|= 0. The open ball with center xand radius δrin (X, dE)is denoted by BdE(x, δr). For any y∈BdE(x, δr), we have that [x, y]⊂[x, ¯x]∪[¯x, y]where ¯x∈[0, x]with dE(x, ¯x) = δr. Then the above estimate gives that (d(x, y) = ´[x,y]λ(t)dt < 2´|x|+δr |x|−δrλ(t)dt < r if |x| ∈ (a, b), x 6= 0, d(x, y) = ´[x,y]λ(t)dt < 2´|x|+δr |x|< r if |x|= 0, and hence y∈Bd(x, r). As Bd(x, r)is arbitrary, we obtain that for any open ball Bd(x, r)there exists δr>0only depending on x, r such that BdE(x, δr)⊂Bd(x, r). Thus (2.1) the identity mapping f: (X, dE)→(X, d)is continuous. Next, we claim that also the identity mapping g: (X, d)→(X, dE),g(x) = xif x∈X, is continuous. Let BdE(x, r′)be an arbitrary open ball with center xand radius r′>0in (X, dE). We set (2.2) δr′= min (ˆ|x| |x|−r′/3 λ(t)dt, ˆ|x|+r′/3 |x| λ(t)dt). Then δr′>0since λ > 0. We denote by Bd(x, δr′)the open ball with center xand radius δr′in (X, d). For any y∈Bd(x, δr′), we have that (2.3) ˆ[x,y] λ(t)dt =d(x, y)< δr′. It follows from (2.2) and (2.3) that |z| ∈ [|x| − r′/3,|x|+r′/3] for any z∈[x, y], and hence dE(x, z)< r′for any z∈[x, y]. In particular, dE(x, y)< r′for any y∈Bd(x, δr′). Then Bd(x, δr′)⊂BdE(x, r′)for any BdE(x, r′). Therefore (2.4) the identity mapping g: (X, d)→(X, dE)is continuous. We conclude from (2.1) and (2.4) that Id X: (X, dE)→(X, d)is a homeomorphism. The claim follows.  We note that Xnis compact in (X, dE)for each n∈Nbecause it is a union of finitely many compact edges. Furthermore, any compact set in (X, dE)is contained in Xnfor some nsince any compact set in (X, dE)is bounded. Since compactness is preserved under homeomorphisms, we have the following corollaries. Corollary 2.2. Let Obe an arbitrary compact set in (X, d). Then O⊂Xnfor some n∈N. Corollary 2.3. Let n∈N. Then Xnis compact in (X, d), and int(Xn)is open in (X, d). Classification criteria for regular trees 9 Corollary 2.4. (X, d, µ)is a connected, locally compact, and non-compact metric measure space. 2.2. Newtonian spaces. Let 1< p < ∞and Xbe a K-regular tree with metric dand measure µas in Section 2.1. Let u∈L1 loc(X). We say that a Borel function g:X→[0,∞]is an upper gradient of uif (2.5) |u(y)−u(z)| ≤ ˆγ g ds whenever y, z ∈Xand γis the geodesic from yto z. In the setting of our tree, any rectifiable curve with end points zand ycontains the geodesic connecting zand y, and therefore the upper gradient defined above is equivalent to the definition which requires that (2.5) holds for all rectifiable curves with end points zand y. In [8, 11], the notion of a p-weak upper gradient is given. A Borel function g:X→[0,∞]is called a p-weak upper gradient of uif (2.5) holds on p-a.e. curve. Here we say that a property holds for p-a.e. curve if it fails only for a curve family Γwith zero p-modulus, i.e., there is a Borel nonnegative function ρ∈Lp(X)such that ´γρ ds =∞for any curve γ∈Γ. We refer to [8, 11] for more information about p-weak upper gradients. The notion of upper gradients is due to Heinonen and Koskela [10], we refer interested readers to [2, 8, 11, 24] for a more detailed discussion on upper gradients. The following lemma of Fuglede shows that a converging sequence in Lphas a subsequence that converges with respect to p-a.e. curve (see [11, Section 5.2]). Lemma 2.5. (Fuglede’s lemma) Let {gn}∞ n=1 be a sequence of Borel nonnegative functions that converges to gin Lp(X). Then there is a subsequence {gnk}∞ k=1 such that lim k→∞ ˆγ |gnk−g|ds = 0 for p-a.e. curve γin X. The following useful results are from [11, Section 2.3 and Section 2.4] or [2, Section 6.1]. Theorem 2.6. Every bounded sequence {un}∞ n=1 in a reflexive normed space (V, |.|V)has a weakly convergent subsequence {unk}∞ k=1. Moreover, there exists u∈V such that unk→uweakly in Vas k→ ∞ and |u|V≤lim inf k→∞ |unk|V. Lemma 2.7. (Mazur’s lemma) Let {un}∞ n=1 be a sequence in a normed space V converging weakly to an element u∈V. Then there exists a sequence ¯vkof convex combinations ¯vk= Nk X i=k λi,kui, Nk X i=k λi,k = 1, λi,k ≥0 converging to vin the norm. The Newtonian space N1,p(X),1< p < ∞, is defined as the collection of all the functions uwith finite N1,p-norm kukN1,p(X):= kukLp(X)+ inf gkgkLp(X) where the infimum is taken over all upper gradients of u. We denote by guthe minimal upper gradient, which is unique up to measure zero and which is minimal in the sense that if g∈Lp(X)is any upper gradient of uthen gu≤ga.e. We refer to 10 Khanh Nguyen [8, Theorem 7.16] for proofs of the existence and uniqueness of such a minimal upper gradient. If u∈N1,p(X), then it is continuous by (2.5) under the assumption λp µ∈ L 1 p−1 loc ([0,∞)) and it has a minimal p-weak upper gradient, see [21, Section 2]. More precisely, by [21, Proposition 2.2] the empty family is the only curve family with zero p-modulus, and hence any p-weak upper gradient is actually an upper gradient here and the conclusion of Lemma 2.5 holds for every curve γ. Moreover, it follows from [8, Definition 7.2 and Lemma 7.6] that any function u∈L1 loc(X)with an upper gradient 0≤g∈Lp(X)is locally absolutely continuous, for example, absolutely continuous on each edge. The classical derivative u′of this locally absolutely continuous function is a minimal upper gradient in the sense that gu=|u′(x)|/λ(x)when uis parametrized in the natural way. We define the homogeneous Newtonian space ˙ N1,p(X),1< p < ∞, the collection of all the continuous functions uthat have an upper gradient 0≤g∈Lp(X), for which the homogeneous ˙ N1,p-norm of udefined as kuk˙ N1,p(X):= |u(0)|+ inf gkgkLp(X) is finite. Here 0is the root of our K-regular tree Xand the infimum is taken over all upper gradients of u. The completion of the family of functions with compact support in N1,p(X)(or ˙ N1,p(X)) is denoted by N1,p 0(X)(or ˙ N1,p 0(X)). We denote by N1,p loc (X)the space of all functions u∈Lp loc(X)that have an upper gradient in Lp loc(X), where Lp loc(X)is the space of all measurable functions that are p-integrable on any compact subset of X. Especially, since each Xnis compact in (X, d)by Corollary 2.3, we conclude that each u∈N1,p loc (X)is both continuous and bounded on each Xn. 2.3. p-(super)harmonic functions. Let 1< p < ∞and Xbe a K-regular tree with metric dand measure µas in Section 2.1. For an open subset Ωof X, a function u∈N1,p loc (Ω) is said to be a p-harmonic function on Ωif (2.6) ˆspt(ϕ) gp udµ ≤ˆspt(ϕ) gp u+ϕdµ holds for all functions ϕ∈N1,p(Ω) with compact support spt(ϕ)⊂Ω. We say that a function u∈N1,p loc (Ω) is a p-superharmonic function if (2.6) holds for all nonnegative functions ϕ∈N1,p(Ω) with compact support spt(ϕ)⊂Ω. We have a characterization of p-harmonic functions on X, see [2, Lemma 7.11]. Theorem 2.8. A function uis a p-harmonic function on Xif and only if uis p-harmonic on int(Xn)for all n∈N. By the stability properties of p-superharmonic functions (superminimizers) in general metric measure spaces (see for instance [2, Theorem 7.25]), since int(Xn)is open for each n∈N(see Corollary 2.3), we obtain the following results in our setting. Theorem 2.9. Let n∈N. If {ui}i≥nis a sequence of p-harmonic functions on int(Xn)which converges locally uniformly to uin int(Xn), then uis p-harmonic on int(Xn). Let 1< p < ∞and n∈N. Then (int(Xn), d, µ)is said to be doubling and to support a p-Poincaré inequality if there exist constants C1≥1, C2>0only Classification criteria for regular trees 17 Proof. We first prove that there exists a nonnegative bounded p-harmonic function uon X. Towards this, Lemma 3.5 and Lemma 3.6 give that there exist constants 0< M2≤M1<∞and a sequence {un}∞ n=1 in N1,p loc (X)with un|En≡1,un|Fn≡0, 0≤un≤1such that unis a nonconstant p-harmonic function on int(Xn)and (3.27) 0< M2≤ˆXn gp undµ =Capp(En, Fn)≤M1<∞for all n∈N. Let n0∈Nbe arbitrary. It follows from the local Hölder continuity of p-harmonic functions (see Theorem 2.10) that {un}n≥n0is equibounded and locally equicontinuous on int(Xn0+1). By the Arzelà-Ascoli theorem, there exists a subsequence, still denoted {un}n≥n0, that converges to uuniformly on Xn0as n→ ∞. Since n0is arbitrary, by uniqueness of locally uniform convergence, we may assume that un converges to ulocally uniformly in Xas n→ ∞. Then {ui}i≥nis a sequence of p-harmonic functions on int(Xn)which converge uniformly to uin int(Xn)for each n, and hence by Theorem 2.9 and Theorem 2.8 we obtain that uis a p-harmonic function on X. Thus uis a nonnegative bounded p-harmonic function on Xsince 0≤un≤1. We next show that 0<ˆX gp udµ < ∞. It follows from (3.27) that {gun}∞ n=1 is a bounded sequence in the reflexive space Lp(X), and hence Theorem 2.6 and Mazur’s Lemma 2.7 give that there exists g∈ Lp(X)and a convex combination sequence ¯gn=PNn i=nai,nguiwith ai,n ≥0,PNn i=nai,n = 1 such that ¯gn→gin Lp(X)as n→ ∞. By Fuglede’s Lemma 2.5, we obtain that there is a subsequence, still denoted ¯gn, such that lim n→∞ ˆ[x,y] ¯gnds =ˆ[x,y] g ds for every curve [x, y]. Note that ¯gn=PNn i=nai,nguiis an upper gradient of ¯un= PNn i=nai,nuiand ¯unconverges to ulocally uniformly in Xas n→ ∞. Hence |u(x)−u(y)|= lim n→∞ |¯un(x)−¯un(y)| ≤ lim n→∞ ˆ[x,y] ¯gnds =ˆ[x,y] g ds for every curve [x, y]. Then gis an upper gradient of uand hence gu≤ga.e. Combining this with ¯gn→gin Lp(X)as n→ ∞ and with the convexity of the function t7→ tpyields (3.28) ˆX gp udµ ≤ˆX gpdµ = lim n→∞ ˆX ¯gp ndµ ≤lim n→∞ Nn X i=n ai,n ˆX gp uidµ. Notice that {Capp(En, Fn)}∞ n=1 is a nonincreasing sequence by Lemma 3.5. Hence we have by ui|Fi≡0, ui|Ei≡1and (3.27) that lim n→∞ Nn X i=n ai,n ˆX gp uidµ = lim n→∞ Nn X i=n ai,n ˆXi gp uidµ = lim n→∞ Nn X i=n ai,nCapp(Ei, Fi) ≤lim n→∞ Nn X i=n ai,nCapp(En, Fn) = lim n→∞ Capp(En, Fn)<∞. Substituting the above estimate into (3.28) yields ´Xgp udµ < ∞. 18 Khanh Nguyen It remains to show that ´Xgp udµ > 0. The preceding being understood, we argue by contradiction and assume that ´Xgp udµ = 0. Then uis constant on X. Recall that ¯gn=PNn i=nai,nguiwith ai,n ≥0,PNn i=nai,n = 1 are such that ¯gn→gin Lp(X)as n→ ∞ and (3.29) lim n→∞ ˆ[x,y] ¯gnds =ˆ[x,y] g ds for every curve [x, y]. Moreover, ¯gnis an upper gradient of ¯un=PNn i=nai,nuiand ¯un is admissible for computing the capacity Capp(ENn, FNn), and so ˆX ¯gp ndµ ≥ˆX gp ¯undµ ≥Capp(ENn, FNn). Combining this with (3.27) and using ¯gn→gin Lp(X)as n→ ∞ yields (3.30) ˆX gpdµ > 0. Let ε > 0be arbitrary, and let [x, y]be an arbitrary edge in X. Since unconverges to ulocally uniformly in X, there exist positive constants N, r only depending on x, y such that for all n≥N, sup t∈B(x,r) |un(t)−u(t)|< ε, sup t∈B(y,r) |un(t)−u(t)|< ε where B(x, r), B(y, r)are balls with centers x, y and radius r, respectively. Since u is constant, the above estimates yield that for all n≥N, |un(x)−un(y)| ≤ |un(x)−u(x)|+|u(y)−un(y)|<2ε. By Section 2.2, unis absolutely continuous on [x, y]and gun(z) = |u′ n(z)|/λ(z)for z∈[x, y]. By the strong maximum principle (see for instance [19, Corollary 6.5]), un is a monotone function on [x, y]. Hence |un(x)−un(y)|=ˆ[x,y] |u′ n(z)|dz. Since the minimal upper gradient gunof unsatisfies gun(x) = |u′ n(x)|/λ(z), we conclude that ˆ[x,y] gunds ≤2ε for all n≥N. By (3.29), it follows that ˆ[x,y] g ds = lim n→∞ ˆ[x,y] ¯gnds = lim n→∞ Nn X i=n ai,n ˆ[x,y] guids < 2ε. Letting ε→0, we obtain that g= 0 a.e. on [x, y]. As [x, y]is arbitrary, we conclude that g= 0 a.e. which contradicts (3.30). This completes the proof.  Proof of Theorem 1.1. Xis p-parabolic ⇔2.is given by Lemma 3.1. 1.⇔2.⇔3.is given by Lemma 3.2.  Proof of Theorem 1.3. Xis p-parabolic ⇒1.is given by Lemma 3.3. 1.⇒2.is trivial. 2.⇒3.: Let ube a bounded p-harmonic function on X. Then there exists a constant C > 0such that u+Cis a nonnegative p-harmonic function on X. Hence u+Cis constant by the assumption and so uis constant. 3.⇒4.is trivial. Classification criteria for regular trees 19 4.⇒Xis p-parabolic is given by Lemma 3.7.  The following example shows that Theorem 1.1 is not true for general metrics and measures. Example 3.8. Let 1< p < ∞. There exists a p-hyperbolic K-regular tree X with a distance and a “non-radial” measure such that Rp=∞. Let us begin with some notation. For simplicity, let Xbe a dyadic tree (which means K= 2). Then the root 0of our tree has two closest vertices, denoted v1and v2. We denote T1= [0, v1]∪ {x∈X:v1∈[0, x]}and T2= [0, v2]∪ {x∈X:v2∈[0, x]}. Note that the union of T1and T2is our tree. Suppose λi, µi: [0,∞)→(0,∞)satisfy λi, µi∈L1 loc([0,∞)), for i= 1,2. We introduce a measure µand a metric dvia ds by setting dµ(x) = µi(|x|)d|x|, ds(x) = λi(|x|)d|x|, for all x∈Ti, for i= 1,2. To obtain what we desire, we choose λ1≡µ1≡1and λ2≡1, µ2(x) = 2−j(x). Define a metric dand a measure µas above. Then Rp|T1:= 1 2ˆ∞ 0 λ p p−1 1(t)µ 1 1−p 1(t)2 j(t) 1−pdt < ∞, Rp|T2:= 1 2ˆ∞ 0 λ p p−1 2(t)µ 1 1−p 2(t)2 j(t) 1−pdt =∞, and Rp=Rp|T1+Rp|T2=∞. By Theorem 1.1 for the subtree T1with Rp|T1<∞, we obtain that T1is p-hyperbolic. Hence there exists a compact set Oin T1such that CapT1 p(O)>0, where CapT1 p(O) := inf ˆT1 gp udµ:u|O≡1, u ∈N1,p 0(T1). Let Obe such a set. Then Ois bounded in T1by Corollary 2.2. Let u∈N1,p 0(X) be an arbitrary function with u|O≡1. It follows from u∈N1,p 0(X)that there exists a sequence un∈N1,p(X)with compact support spt(un)⊂Xsuch that un→uin N1,p(X)as n→ ∞. By Corollary 2.2, we may assume that spt(un)⊂Xnfor each n. Hence spt(un)∩T1⊂Xn∩T1, and so spt(un)∩T1is compact in T1because spt(un)∩T1is a closed set in T1and Xn∩T1is compact in T1. Then for each n, un∈N1,p(T1)with compact support spt(un)and un→uin N1,p(T1)as n→ ∞, and hence u∈N1,p 0(T1)with u|O≡1. Thus uis admissible for computing CapT1 p(O)and so ˆX gp udµ ≥ˆT1 gp udµ ≥CapT1 p(O). Since u∈N1,p 0(X)with u|O≡1is arbitrary, the above estimate gives Capp(O)≥CapT1 p(O). Combining this with CapT1 p(O)>0, we have Capp(O)>0. Thus Xis p-hyperbolic. Example 3.9. Let 1< p < ∞. There exist both p-hyperbolic and p-parabolic K-regular trees (X, d, µ)that are doubling and support a p-Poincaré inequality. 20 Khanh Nguyen We begin with the p-hyperbolic case. Let µ(t) = e−βj(t)and λ(t) = e−εj(t)with ε, β > 0and log K < β < log K+εp. It is obvious that µ(X)<∞and Rp<∞. More precisely, since log K < β < log K+εp we have that µ(X) = ˆ∞ 0 µ(t)Kj(t)dt =ˆ∞ 0 e−(β−log K)j(t)dt < ∞ and Rp=ˆ∞ 0 λ(t)p p−1µ(t)1 1−pKj(t) 1−pdt =ˆ∞ 0 e(β−log K−εp)j(t) p−1dt < ∞. 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[28] Yamasaki, M.: Parabolic and hyperbolic infinite networks. - Hiroshima Math. J. 7:1, 1977, 135–146. Received 5 November 2020 •Accepted 12 March 2021 •Published online 29 November 2021 Khanh Nguyen University of Jyväskylä Department of Mathematics and Statistics P. O. Box 35 FI-40014 University of Jyväskylä Finland khanh.n.nguye[email protected]