aa r X i v : . [ m a t h . M G ] S e p Classification criteria for regular trees
Khanh Ngoc Nguyen ∗ Abstract
We give characterizations for the parabolicity of regular trees.
Let us begin with the uniformization theorem of F. Klein, P. Koebe and H. Poincar´efor Riemann surfaces. The celebrated theorem says that every simply connected Riemannsurface M is conformally equivalent (or bi-holomorphic) to one of three Riemann surfaces:the half plane H (surface of hyperbolic type), the Euclidean plane R (surface of parabolictype), the unit sphere S (surface of elliptic type). Then M admits a Riemannian metric g with constant curvature. A simply connected Riemann surface is said to be hyperbolic if it is conformally equivalent to H , otherwise we say that it is parabolic .Let M be a simply connected Riemann surface with Riemannian metric g . A C -smooth function u defined in M is superharmonic if∆ u ≤ g .It is well known that every conformal mapping in dimension two preserves super-harmonic functions (see [1, Page 135]). Since H possesses a nonconstant nonnegativesuperharmonic function and every nonnegative superharmonic function on R or S is con-stant, it then follows that there is no nonconstant nonnegative superharmonic functionon ( M, g ) if and only if M is parabolic.Let K be a compact subset in ( M, g ). We define the capacity
Cap( K ) byCap( K ) = inf (cid:26)Z M |∇ u | dm g : u ∈ Lip ( M ) , u | K ≡ (cid:27) Mathematics Subject classfication : 31C05, 31C15, 31C45, 31E05
Key words and phases : capacity, harmonic function, parabolicity, regular treeThe author has been supported by the Academy of Finland (project No. 323960) ∗ Department of Mathematics and Statistics, University of Jyv¨askyl¨a, PO Box 35, FI-40014 Jyv¨askyl¨a,Finland. E-mail address: khanh.n.nguyen@jyu.fi, [email protected] Khanh Ngoc Nguyen where Lip ( M ) is a set of all locally Lipschitz functions with a compact support on M , and m g is the Riemannian measure associated to g . Then there is a nonconstant nonnegativesuperharmonic function on M if and only if Cap( K ) > K ,(see [7, Theorem 5.1] for Riemannian manifolds). It follows that the parabolicity of aRiemann surface M can be characterized both in terms of capacity and superharmonicfunctions. By this reason, in the setting of Riemannian manifolds or metric measurespaces, one defines parabolicity either via capacity (see [12, 14–16]) or via superharmonicfunctions (see [7] and also references therein). In this paper, we will consider K -regulartrees and give the definition of parabolicity in terms of capacity.Recently, analysis on K -regular trees has been under development, see [4, 20–23, 27].Let G be a rooted K -ary tree with a set of vertices V and a set of edges E for some K ≥
1. The union of V and E will be denoted by X . We abuse the notation and call X a K -ary tree or a K -regular tree. We introduce a metric structure on X by considering eachedge of X to be an isometric copy of the unit interval. Then the distance between twovertices is the number of edges needed to connect them and there is a unique geodesic thatminimizes this number. Let us denote the root by 0. If x is a vertex, we define | x | to bethe distance between 0 and x . Since each edge is an isometric copy of the unit interval, wemay extend this distance naturally to any x belonging to an edge. We refer the interestedreaders to [20] for a discussion on a definition of K -regular trees based on sets in the plane R . We define ∂X as the collection of all infinite geodesics starting at the root 0. Thenevery ξ ∈ ∂X corresponds to an infinite geodesic [0 , ξ ) (in X ) that is an isometric copy ofthe interval [0 , ∞ ). Let µ and λ : [0 , ∞ ) → (0 , ∞ ) be locally integrable functions. Let d | x | be the length element on X . We define a measure µ on X by setting dµ ( x ) = µ ( | x | ) d | x | ,and a metric d on X via ds ( x ) = λ ( | x | ) d | x | by setting d ( y, z ) = R [ y,z ] ds whenever y, z ∈ X and [ y, z ] is the unique geodesic between y and z . Then ( X, d, µ ) is a metric measurespace and hence one may define a Newtonian Sobolev space N ,p ( X ) := N ,p ( X, d, µ )based on upper gradients [10, 24]. As usual, N ,p ( X ) is the completion of the family offunctions with compact support in N ,p ( X ), and ˙ N ,p ( X ) is the completion of the familyof functions with compact support in ˙ N ,p ( X ), the homogeneous version of N ,p ( X ). LetΩ be a subset of X . We denote by N ,p loc (Ω) the space of all functions u ∈ L p loc (Ω) that havean upper gradient in L p loc (Ω), where L p loc (Ω) is the space of all measurable functions thatare p -integrable on any compact subset of Ω. See Section 2 for the precise definitions.Let 1 < p < ∞ and O be a subset of X . We define the p -capacity of O , denotedCap p ( O ), by setting(1.1) Cap p ( O ) = inf (cid:26)Z X g pu dµ : u | O ≡ , u ∈ N ,p ( X ) (cid:27) where g u is the minimal upper gradient of u as in Section 2.2. A K -regular tree X is saidto be p -parabolic if Cap p ( O ) = 0 for all compact sets O ⊂ X ; otherwise X is p -hyperbolic .Given 1 < p < ∞ and a subset Ω ⊆ X , we say that u ∈ N ,p loc (Ω) is a p -harmonic lassification criteria for regular trees function (or a p -superharmonic function ) on Ω if(1.2) Z spt( ϕ ) g pu dµ ≤ Z spt( ϕ ) g pu + ϕ dµ holds for all functions (or for all nonnegative functions) ϕ ∈ N ,p (Ω) with compact supportspt( ϕ ) ⊂ Ω. We refer the interested readers to [3, 9, 11] for a discussion on the p -capacityand p -(super)harmonic functions.Since a K -regular tree ( X, d ) is the quintessential Gromov hyperbolic space, it is thennatural to ask for whether the parabolicity (or hyperbolicity) of X can be characterizedvia p -(super)harmonic functions under some conditions on the measure µ only dependingon the given metric d , and also ask for intrinsic conditions of K -regular trees that wouldcharacterize the parabolicity (or hyperbolicity). We refer the readers to [1, Chapter IV] fora discussion in the case of Riemann surfaces, and [6,7,14–16] for a discussion in the settingof Riemannian manifolds, and [25, Section 6], [26] for a discussion on infinite networks.In order to state our results, we introduce a notion from [21]. Let 1 < p < ∞ . We set R p ( λ, µ ) = Z ∞ λ ( t ) pp − µ ( t ) − p K j ( t )1 − p dt where j ( t ) is the smallest integer such that j ( t ) ≥ t , and let X n = { x ∈ X : | x | ≤ n } for each n ∈ N . Since we work with a fixed pair λ, µ , we will usually write R p ( λ, µ )simply as R p when no confusion can arise. In what follows, we additionally assume that λ p µ − ∈ L / ( p − ([0 , ∞ )) to make sure that the finiteness of R p is a condition at infinity.The first result of our paper is a characterization of parabolicity of K -regular trees. Theorem 1.1.
Let < p < ∞ and X be a K -regular tree with metric d and measure µ as above. Then ( X, d, µ ) is p-parabolic if and only if any one of the following conditionsis fulfilled:1. R p ( λ, µ ) = ∞ .2. Cap p ( X n ) = 0 for all n ∈ N ∪ { } .3. Cap p ( X n ) = 0 for some n ∈ N ∪ { } . In Section 2.1, we will show that the compactness on a K -regular tree X with respectto our metric d and with respect to the graph metric are equivalent. Since each compactset in ( X, d ) is contained in some n -level set X n that is an analog of a ball with respectto graph metric, parabolicity of X can be characterized by the zero p -capacity of some/all n -level sets X n .In [21, Theorem 1.3], the condition R p ( λ, µ ) = ∞ gives a characterization of theexistence of boundary trace operators and for density properties for ˙ N ,p ( X ). Henceparabolicity of K -regular trees can be characterized in terms of boundary trace operatorsand density properties. Combining Theorem 1.1 and [21, Theorem 1.3 and Theorem 3.5],we obtain the following corollary. Khanh Ngoc Nguyen
Corollary 1.2.
Let < p < ∞ and X be a K -regular tree with metric d and measure µ as above. Then ( X, d, µ ) is p -parabolic if and only if any one of the following conditionsis fulfilled:1. There exists u ∈ ˙ N ,p ( X ) such that lim [0 ,ξ ) ∋ x → ξ u ( x ) = ∞ for all ξ ∈ ∂X .2. ˙ N ,p ( X ) = ˙ N ,p ( X ) . It is well known, see for instance the survey paper [15], that the volume growth condi-tion Z ∞ (cid:18) tV ( B (0 , t )) (cid:19) p − dt = ∞ is a sufficient condition to guarantee parabolicity of Riemannian manifolds. Here V ( B (0 , t ))is the volume of the ball with radius t and center at a fixed point 0. However, this con-dition is far from being necessary in general, as shown by a counterexample due to I.Holopainen [14]. Our condition R p ( λ, µ ) = ∞ is an analog of this volume growth condi-tion. Example 3.8 in Section 3 shows that there exists a K -regular tree with a distance d and a “non-radial” measure µ such that R p ( λ, µ ) = ∞ but X is p -hyperbolic.Let 1 < p < ∞ and let Ω be a subset of X . We say that (Ω , d, µ ) is doubling andsupports a p -Poincar´e inequality if there exist constants C ≥ , C >
0, and τ ≥ B ( x, r ) ⊂ Ω, µ ( B ( x, r )) ≤ C µ ( B ( x, r ))and for all balls B ( x, τ r ) ⊂ Ω, − Z B ( x,r ) | u − u B ( x,r ) | dµ ≤ C r (cid:18) − Z B ( x,τr ) g p dµ (cid:19) p whenever u is a measurable function on B ( x, τ r ) and g is an upper gradient of u , where u B ( x,r ) := − R B ( x,r ) udµ = µ ( B ( x,r )) R B ( x,r ) udµ. The validity of p -Poincar´e inequality for X has very recently been characterized via a Muckenhoupt-type condition under a doublingcondition 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 and Green’s functions. The definition of Green’s functions is given inSection 2.3. Theorem 1.3.
Let < p < ∞ and X be a K -regular tree with metric d and measure µ as above. Assume additionally that ( X n , d, µ ) is doubling and supports a p -Poincar´einequality for each n ∈ N . Then ( X, d, µ ) is p-parabolic if and only if any one of thefollowing conditions is fulfilled: lassification criteria for regular trees
1. Every nonnegative p -superharmonic function u on X is constant.2. Every nonnegative p -harmonic function u on X is constant.3. Every bounded p -harmonic function u on X is constant.4. Every p -harmonic function u on X with R X g pu dµ < ∞ is constant.5. Every bounded p -harmonic function u on X with R X g pu dµ < ∞ is constant.6. There is no Green’s function on X . Let us close the introduction with some comments on Theorem 1.3. According to aversion of Theorem 1.3 in the setting of Riemannian manifolds from [12, 13, 17] we havethat { ., . } ⇒ . ⇒ . ⇒ . ⇔ . . However 6 . : . : . : . in general. Fur-thermore, if K = 1 then Theorem 1.3 gives a characterization of parabolicity in termsof p -(super)harmonic functions and Green’s functions on the half line R + . By the com-pactness properties of a K -regular tree in Section 2.1, our condition that ( X n , d, µ ) isdoubling and supports a p -Poincar´e inequality for each n ∈ N is equivalent to µ being alocally doubling measure supporting a local p -Poincar´e inequality on ( X, d ), i.e (Ω , d, µ )is doubling and supports a p -Poincar´e inequality for any compact subset Ω in ( 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 -Poincar´e inequality. SeeExample 3.9 in Section 3 for more details.The motivation for our paper comes from classification problems of spaces. By thesurvey papers [2, 7], the development of potential theory in the setting of metric measurespaces leads to a classification of spaces as either p -parabolic or not. This dichotomy canbe seen as a non-linear analog of the recurrence or transience dichotomy in the theory ofBrownian motion. This classification is helpful in the development of a quasiconformaluniformization theory, or for a deeper understanding of the links between the geometryof hyperbolic spaces and the analysis on their boundaries at infinity.The paper is organized as follows. In Section 2, we introduce K -regular trees, Newto-nian spaces, Green’s functions, and p -(super)harmonic functions on our trees. In Section3, we give the proofs of Theorem 1.1 and Theorem 1.3.Throughout this paper, the letter C (sometimes with a subscript) will denote positiveconstants that usually depend only on the space and may change at different occurrences;if C depends on a, b, . . . , we write C = C ( a, b, . . . ). For any function f ∈ L ( X ) and anymeasurable subset A ⊂ X , let − R A f dµ stand for µ ( A ) R A f dµ . Khanh Ngoc Nguyen A graph G is a pair ( V, E ), where V is a set of vertices and E is a set of edges. Wecall a pair of vertices x, y ∈ V neighbors if x is connected to y by an edge. The degreeof a vertex is the number of its neighbors. The graph structure gives rise to a naturalconnectivity structure. A tree G is 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 G a rooted tree if it has a distinguished vertex called the root , which wewill denote by 0. The neighbors of a vertex x ∈ V are of two types: the neighbors thatare closer to the root are called parents of x and all other neighbors are called children of x . Each vertex has a unique parent, except for the root itself that has none.A K -ary tree G is a rooted tree such that each vertex has exactly K children. Thenall vertices except the root of a K -ary tree have degree K + 1, and the root has degree K .In this paper, we say that a tree is K -regular if it is a K -ary tree for some integer K ≥ G be a K -regular tree with a set of vertices V and a set of edges E for someinteger K ≥
1. For simplicity of notation, we let X = V ∪ E and call it a K -regulartree. For x ∈ X , let | x | be the distance from the root 0 to x , that is, the length of thegeodesic from 0 to x , where the length of every edge is 1 and we consider each edge tobe an isometric copy of the unit interval. The geodesic connecting two points x, y ∈ X isdenoted by [ x, y ]. More precisely, we refer the interested readers to [20] for a discussionon a definition of K -regular trees in the plane R , that is equivalent to the notions of our K -regular trees.On our K -regular tree X , we define a measure µ and a metric d via 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 ∈ L ([0 , ∞ )). Here d | x | is the measurewhich gives each edge Lebesgue measure 1, as we consider each edge to be an isometriccopy of the unit interval and the vertices are the end points of this interval. Hence forany two points z, y ∈ X , the distance between them is d ( z, y ) = Z [ z,y ] ds ( x ) = Z [ z,y ] λ ( | x | ) d | x | where [ z, y ] is the unique geodesic from z to y in 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 d E the Euclidean metric or the graph metric on X . Then for any twopoints z, y ∈ X , d E ( z, y ) = Z [ z,y ] d | x | lassification criteria for regular trees z and y where [ z, y ] is the uniquegeodesic from z to y . Theorem 2.1.
The identity mapping Id X : ( X, d E ) → ( X, d ) is a homeomorphism.Proof. Let us first prove that the identity mapping f : ( X, d E ) → ( X, d ), f ( x ) = x if x ∈ X , is continuous. Let B d ( x, r ) be an arbitrary open ball with center x and radius r > X, d ). Recall that λ : [0 , ∞ ) → (0 , ∞ ) is a locally integrable function. Hence λ is a integrable function on [ a, b ] wherever [ a, b ] is a compact interval with | x | ∈ ( a, b ) if x = 0, or | x | = a if x = 0 where 0 is the root of X . Then F ( h ) := Z ha λ ( t ) dt is absolutely continuous on [ a, b ]. It follows that there exists δ r > x, r such that (R | x | + δ r | x |− δ r λ ( t ) dt < r if | x | ∈ ( a, b ) , x = 0 , R | x | + δ r | x | λ ( t ) dt < r if | x | = 0 . The open ball with center x and radius δ r in ( X, d E ) is denoted by B d E ( x, δ r ). For any y ∈ B d E ( x, δ r ), we have that [ x, y ] ⊂ [ x, ¯ x ] ∪ [¯ x, y ] where ¯ x ∈ [0 , x ] with d E ( x, ¯ x ) = δ r .Then the above estimate gives that ( d ( x, y ) = R [ x,y ] λ ( t ) dt < R | x | + δ r | x |− δ r λ ( t ) dt < r if | x | ∈ ( a, b ) , x = 0 ,d ( x, y ) = R [ x,y ] λ ( t ) dt < R | x | + δ r | x | < r if | x | = 0 , and hence y ∈ B d ( x, r ). As B d ( x, r ) is arbitrary, we obtain that for any open ball B d ( x, r )there exists δ r > x, r such that B d E ( x, δ r ) ⊂ B d ( x, r ) . Thus(2.1) the identity mapping f : ( X, d E ) → ( X, d ) is continuous . Next, we claim that also the identity mapping g : ( X, d ) → ( X, d E ), g ( x ) = x if x ∈ X ,is continuous. Let B d E ( x, r ′ ) be an arbitrary open ball with center x and radius r ′ > X, d E ). We set(2.2) δ r ′ = min (Z | x || x |− r ′ / λ ( t ) dt, Z | x | + r ′ / | x | λ ( t ) dt ) . Let [ a, b ] be an arbitrary closed subinterval of [0 , ∞ ) with a < b . Note that λ is strictlypositive on [0 , ∞ ). Then [ a, b ] = ∪ ∞ k =1 A k where A k = { t ∈ [ a, b ] : λ ( t ) > k } , and so theremust be a k such that L ( A k ) > L is the Lebesgue measure on [0 , ∞ ). We havefrom A k ⊂ [ a, b ] and L ( A k ) > Z [ a,b ] λ ( t ) dt ≥ Z A k λ ( t ) dt ≥ k L ( A k ) > Khanh Ngoc Nguyen for any [ a, b ] with 0 ≤ a < b < ∞ . Combining this with (2.2) yields δ r ′ >
0. We denoteby B d ( x, δ r ′ ) the open ball with center x and radius δ r ′ in ( X, d ). For any y ∈ B d ( x, δ r ′ ),we have that(2.4) Z [ x,y ] λ ( t ) dt = d ( x, y ) < δ r ′ . It follows from (2.2) and(2.4) that | z | ∈ [ | x | − r ′ / , | x | + r ′ /
2] for any z ∈ [ x, y ], and hence d E ( x, z ) < r ′ for any z ∈ [ x, y ]. In particular, d E ( x, y ) < r ′ for any y ∈ B d ( x, δ r ′ ). Then B d ( x, δ r ′ ) ⊂ B d E ( x, r ′ ) for any B d E ( x, r ′ ). Therefore(2.5) the identity mapping g : ( X, d ) → ( X, d E ) is continuous.We conclude from (2.1) and (2.5) that Id X : ( X, d E ) → ( X, d ) is a homeomorphism. Theclaim follows.We note that X n is compact in ( X, d E ) for each n ∈ N because it is a union of finitelymany compact edges. Furthermore, any compact set in ( X, d E ) is contained in X n forsome n since any compact set in ( X, d E ) is bounded. Since compactness is preservedunder homeomorphisms, we have the following corollaries. Corollary 2.2.
Let O be an arbitrary compact set in ( X, d ) . Then O ⊂ X n for some n ∈ N . Corollary 2.3.
Let n ∈ N . Then X n is compact in ( X, d ) . Corollary 2.4. ( X, d, µ ) is a connected, locally compact, and non-compact metric measurespace. Let 1 < p < ∞ and X be a K -regular tree with metric d and measure µ as in Section2.1. Let u ∈ L ( X ). We say that a Borel function g : X → [0 , ∞ ] is an upper gradient of u if(2.6) | u ( y ) − u ( z ) | ≤ Z γ gds whenever y, z ∈ X and γ is the geodesic from y to z . In the setting of our tree, anyrectifiable curve with end points z and y contains the geodesic connecting z and y , andtherefore the upper gradient defined above is equivalent to the definition which requiresthat (2.6) holds for all rectifiable curves with end points z and y . In [8, 11], the notionof a p -weak upper gradient is given. A Borel function g : X → [0 , ∞ ] is called a p -weakupper gradient of u if (2.6) 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 Γ with zero p -modulus , i.e., there is a Borel lassification criteria for regular trees ρ ∈ L p ( X ) such that R γ ρ ds = ∞ for any curve γ ∈ Γ. We referto [8, 11] for more information about p -weak upper gradients.The notion of upper gradients is due to Heinonen and Koskela [10], we refer interestedreaders to [3, 8, 11, 24] for a more detailed discussion on upper gradients.The following lemma of Fuglede shows that a convergence sequence in L p has a subse-quence that converges with respect to p -a.e curve (see [11, Section 5.2]). Lemma 2.5 (Fuglede’s lemma) . Let { g n } ∞ n =1 be a sequence of Borel nonnegative functionsthat converges to g in L p ( X ) . Then there is a subsequence { g n k } ∞ k =1 such that lim k →∞ Z γ | g n k − g | ds = 0 for p -a.e curve γ in X . The following useful results are from [11, Section 2.3 and Section 2.4] or [3, Section6.1].
Theorem 2.6.
Every bounded sequence { u n } ∞ n =1 in a reflexive normed space ( V, | . | V ) has aweakly convergent subsequence { u n k } ∞ k =1 . Moreover, there exists u ∈ V such that u n k → u weakly in V as k → ∞ and | u | V ≤ lim inf k →∞ | u n k | V . Lemma 2.7 (Mazur’s lemma) . Let { u n } ∞ n =1 be a sequence in a normed space V convergingweakly to an element u ∈ V . Then there exists a sequence ¯ v k of convex combinations ¯ v k = N k X i = k λ i,k u i , N k X i = k λ i,k = 1 , λ i,k ≥ converging to v in the norm. The
Newtonian space N ,p ( X ), 1 < p < ∞ , is defined as the collection of all thefunctions u with finite N ,p -norm k u k N ,p ( X ) := k u k L p ( X ) + inf g k g k L p ( X ) where the infimum is taken over all upper gradients of u . We denote by g u the minimalupper gradient, which is unique up to measure zero and which is minimal in the sense thatif g ∈ L p ( X ) is any upper gradient of u then g u ≤ g a.e.. We refer to [8, Theorem 7.16]for proofs of the existence and uniqueness of such a minimal upper gradient. Throughoutthis paper, we denote by g u the minimal upper gradient of u .If u ∈ N ,p ( X ), then it is continuous by (2.6) under the assumption λ p /µ ∈ L / ( p − ([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, and0 Khanh Ngoc Nguyen hence any p -weak upper gradient is actually an upper gradient here. Moreover, it fol-lows from [8, Definition 7.2 and Lemma 7.6] that any function u ∈ L ( X ) with anupper gradient 0 ≤ g ∈ L p ( X ) is locally absolutely continuous, for example, absolutelycontinuous on each edge. The ˆa ˘AIJclassicalˆa ˘A˙I derivative u ′ of this locally absolutelycontinuous function is a minimal upper gradient in the sense that g u = | u ′ ( x ) | /λ ( x ) when u is parametrized in the nature way.We define the homogeneous Newtonian spaces ˙ N ,p ( X ), 1 < p < ∞ , the collection ofall the continuous functions u that have an upper gradient 0 ≤ g ∈ L p ( X ), for which thehomogeneous ˙ N ,p -norm of u defined as k u k ˙ N ,p ( X ) := | u (0) | + inf g k g k L p ( X ) is finite. Here 0 is the root of our K -regular tree X and the infimum is taken over allupper gradients of u .The completion of the family of functions with compact support in N ,p ( X ) (or ˙ N ,p ( X ))is denoted by N ,p ( X ) (or ˙ N ,p ( X )). We denote by N ,p loc ( X ) the space of all functions u ∈ L p loc ( X ) that have an upper gradient in L p loc ( X ), where L p loc ( X ) is the space of allmeasurable functions that are p -integrable on any compact subset of X . Especially, sinceeach X n is compact in ( X, d ) by Corollary 2.3, we conclude that each u ∈ N ,p loc ( X ) is bothcontinuous and bounded on each X n . p -(super)harmonic functions Let 1 < p < ∞ and X be a K -regular tree with metric d and measure µ as in Section2.1. For any subset Ω of X , a function u ∈ N ,p loc (Ω) is said to be a p -harmonic functionon Ω if(2.7) Z spt( ϕ ) g pu dµ ≤ Z spt( ϕ ) g pu + ϕ dµ holds for all functions ϕ ∈ N ,p (Ω) with compact support spt( ϕ ) ⊂ Ω. We say thata function u ∈ N ,p loc (Ω) is a p -superharmonic function if (2.7) holds for all nonnegativefunctions ϕ ∈ N ,p (Ω) with compact support spt( ϕ ) ⊂ Ω.We give another definition of p -(super)harmonic functions on a compact set. Thefollowing proposition shows that the notion of a p -(super)harmonic function is the sameas notion of a (super)minimizer in [3, Section 7.3: Definition 7.7 and Proposition 7.9] ona compact set. Proposition 2.8.
Let u ∈ N ,p loc (Ω) and Ω be an arbitrary compact subset in ( X, d ) . Thenthe following are equivalent1. The function u is a p -(super)harmonic function on Ω . lassification criteria for regular trees
2. For all (nonnegative) functions ϕ ∈ N ,p (Ω) we have Z spt( ϕ ) g pu dµ ≤ Z spt( ϕ ) g pu + ϕ dµ. Proof.
It is clear that 2 . ⇒ . since N ,p (Ω) contains all functions in N ,p (Ω) with compactsupport in Ω. Conversely, let ϕ ∈ N ,p (Ω) be an arbitrary (nonnegative) function andlet ε > ϕ ε ∈ N ,p (Ω) withcompact support spt( ϕ ε ) ⊂ Ω such that k ϕ − ϕ ε k N ,p (Ω) < ε. It follows from (2.7) that (cid:18)Z spt( ϕ ε ) g pu dµ (cid:19) /p ≤ (cid:18)Z spt( ϕ ε ) g pu + ϕ ε dµ (cid:19) /p . Let A := { x ∈ Ω : ϕ ε ( x ) = 0 = ϕ ( x ) } and B := { x ∈ Ω : ϕ ε ( x ) = 0 = ϕ ( x ) } . Then spt( ϕ ) =spt( ϕ ε ) ∪ A \ B . As g u = g u + ϕ ε on A , adding (cid:0)R A g pu dµ (cid:1) /p = (cid:0)R A g pu + ϕ ε (cid:1) /p to both sidesof above estimate, using the triangle inequality and k ϕ − ϕ ε k N ,p (Ω) < ε we obtain that (cid:18)Z spt( ϕ ε ) ∪ A g pu dµ (cid:19) /p ≤ (cid:18)Z spt( ϕ ε ) ∪ A g pu + ϕ ε dµ (cid:19) /p ≤ (cid:18)Z spt( ϕ ε ) ∪ A g pu + ϕ dµ (cid:19) /p + ε. Note that R B g pu dµ < ∞ because u ∈ N ,p loc (Ω), B ⊂ Ω, and Ω is compact in (
X, d ). Since g u = g u + ϕ on B , we can subtract (cid:0)R B g pu dµ (cid:1) /p = (cid:0)R B g pu + ϕ dµ (cid:1) /p from both sides of aboveestimate to obtain that (cid:18)Z spt( ϕ ε ) ∪ A \ B g pu dµ (cid:19) /p ≤k g u χ spt( ϕ ε ) ∩ A k L p (Ω) − k g u χ B k L p (Ω) ≤k g u + ϕ χ spt( ϕ ε ) ∩ A k L p (Ω) + ε − k g u + ϕ χ B k L p (Ω) ≤ (cid:18)Z spt( ϕ ε ) ∪ A \ B g pu + ϕ dµ (cid:19) /p + ε. Combining this with spt( ϕ ) = spt( ϕ ε ) ∪ A \ B and letting ε → Z spt( ϕ ) g pu dµ ≤ Z spt( ϕ ) g pu + ϕ dµ for all (nonnegative) functions ϕ ∈ N ,p (Ω). This completes the proof.We next give a characterization of p -superharmonic functions on X . Theorem 2.9.
A function u is a p -superharmonic function on X if and only if u is p -superharmonic on X n for all n ∈ N . Khanh Ngoc Nguyen
Proof.
Assume first that u is a p -superharmonic function on X . Then for each n ∈ N , u is also p -superharmonic on X n since any compact support in X n is also a compactsupport in X . Conversely, suppose that u is p -superharmonic on X n for all n ∈ N .Let 0 ≤ ϕ ∈ N ,p ( X ) be an arbitrary function with compact support spt( ϕ ) ⊂ X . ByCorollary 2.2, spt( ϕ ) ⊂ X n for some n ∈ N . As u is p -superharmonic on X n we have Z spt( ϕ ) g pu dµ ≤ Z spt( ϕ ) g pu + ϕ dµ. Since ϕ is arbitrary, we obtain that u is a p -superharmonic function on X . The claimfollows.By the stability properties of p -superharmonic functions (superminimizers) in generalmetric measure spaces (see for instance [3, Theorem 7.25]) and since X n is compact foreach n ∈ N (see Corollary 2.3), we obtain the following results in our setting. Theorem 2.10.
Let Ω be a compact subset in ( X, d ) . Let { u n } ∞ n =1 be a sequence of p -superharmonic functions on Ω which converges locally uniformly to u in Ω . Then u is a p -superharmonic function on Ω . In particular, if { u i } i ≥ n is a sequence of p -superharmonicfunctions on X n which converges locally uniformly to u in X n then u is p -superharmonicon X n . Let 1 < p < ∞ and Ω be a subset of X . Then (Ω , d, µ ) is said to be doubling and tosupport a p -Poincar´e inequality if there exist constants C ≥ , C > , and τ ≥ B ( x, r ) ⊂ Ω, µ ( B ( x, r )) ≤ C µ ( B ( x, r ))and for all balls B ( x, τ r ) ⊂ Ω, − Z B ( x,r ) | u − u B ( x,r ) | dµ ≤ C r (cid:18) − Z B ( x,τr ) g p dµ (cid:19) p whenever u is a measurable function on B ( x, τ r ) and g is an upper gradient of u . Recallthat if (Ω , d, µ ) is doubling and supports a p -Poincar´e inequality then N ,p (Ω , d, µ ) is areflexive space (see [5, Theorem 4.48]).Combining Proposition 3.9 and Theorem 5.4 in [19], we obtain the local H¨older conti-nuity of p -harmonic functions on X n for each n ∈ N . Theorem 2.11.
Let n ∈ N . Assume that ( X n , d, µ ) is doubling and supports a p -Poincar´einequality. Then every p -harmonic function u on X n is locally α -H¨older continuous forsome < α ≤ . We modify slightly the definitions of p -singular (Green’s) functions in [18, Definition3.11] and define the corresponding Green’s functions of our K -regular trees as follows. lassification criteria for regular trees Definition 2.12.
Let < p < ∞ and X be a K -regular tree with metric d and measure µ as in Section 2.1. A nonconstant extended real-valued function g on X is said to be aGreen’s function on X with singularity at y ∈ X if the following four criteria are met:1. lim x → y g ( x ) = Cap p ( { y } ) − p , where we adopt a convention that Cap p ( { y } ) − p = ∞ if Cap p ( { y } ) = 0 . g > on X and is p -harmonic on X \ { y } .3. For sufficiently small r > , whenever x ∈ S ( y, r ) := { x ∈ X : d ( x, y ) = r } , wehave g ( x ) ≈ lim n →∞ Cap p ( B ( y, r ) , X n ) − p with the comparison constant depending only on the singularity y , where B ( y, r ) = { x ∈ X : d ( x, r ) ≤ r } and Cap p ( B ( y, r ) , X n ) := inf (cid:26)Z X g pu dµ : u ∈ N ,p ( X ) , u | B ( y,r ) ≡ , u | X \ X n ≡ (cid:27) .
4. There exists b > such that for all b with b ≤ b ≤ Cap p ( { y } ) − p and for all a with ≤ a < b < ∞ , we have (cid:18) p − p (cid:19) p − b − a ) p − ≤ Cap p ( X b , X a ) ≤ p ( b − a ) p − where X b := { x ∈ X : g ( x ) ≥ b } , X a := { x ∈ X : g ( x ) > a } , and Cap p ( X b , X a ) := inf (cid:26)Z X g pu dµ : u ∈ N ,p ( X ) , u | X b ≡ , u | X a ≡ (cid:27) . A function g on X is said to be a Green’s function if g is a Green’s function on X withsingularity at y for some y ∈ X . Notice that (
X, d, µ ) is a connected, locally compact, and non-compact metric measurespace by Corollary 2.4. Then we obtain a characterization of parabolicity in terms ofGreen’s functions (see [18, Theorem 3.14]).
Theorem 2.13.
Let n ∈ N . Suppose that ( X n , d, µ ) is doubling and supports a p -Poincar´einequality for each n . Then ( X, d, µ ) is p -parabolic if and only if there is no Green’sfunction on X . Khanh Ngoc Nguyen
In this section, if we do not specifically mention, we always assume that 1 < p < ∞ and that X is a K -regular tree with metric d and measure µ as in Section 2.1. Lemma 3.1. X is p -parabolic if and only if Cap p ( X n ) = 0 for all n ∈ N ∪ { } .Proof. Let X be p -parabolic. By Corollary 2.3, we have that X n is compact in ( X, d ) forall n ∈ N and hence Cap p ( X n ) = 0for all n ∈ N . This also holds for all n ∈ N ∪ { } because Cap p ( X ) ≤ Cap p ( X n ).Conversely, suppose that(3.1) Cap p ( X n ) = 0for all n ∈ N ∪ { } . Let O be an arbitrary compact set in ( X, d ). Then O ⊂ X n for some n ∈ N by Corollary 2.2, and so that Cap p ( O ) ≤ Cap p ( X n ). Combining this with (3.1)yields Cap p ( O ) = 0. Since O is arbitrary, we conclude that X is p -parabolic. The proofis complete. Lemma 3.2.
Let n ∈ N ∪ { } be arbitrary. Then R p = ∞ if and only if Cap p ( X n ) = 0 .Proof. Suppose that R p = ∞ . We first claim that Cap p ( X n ) = 0. Note that λ, µ :[0 , ∞ ) → (0 , ∞ ) are locally integrable functions with λ p µ − ∈ L /p − ([0 , ∞ )). Hence λ pp − ( t ) µ − p ( t ) K j ( t )1 − p > t ∈ [0 , ∞ ). By a similar argument of (2.3), we obtain that(3.2) Z kn λ pp − ( t ) µ − p ( t ) K j ( t )1 − p dt > k ≥ n + 1. Let us define a sequence { u k } ∞ k = n +1 by setting(3.3) u k ( x ) = x ∈ X n , − R | x | n λ ( t ) pp − µ − p ( t ) K j ( t )1 − p dt R kn λ ( t ) pp − µ − p ( t ) K j ( t )1 − p dt if x ∈ X k \ X n , . Then g k ( x ) = λ ( x ) p − µ ( x ) − p K j ( x )1 − p R kn λ ( t ) pp − µ ( t ) − p K j ( t )1 − p dt χ X k \ X n ( x )is an upper gradient of u k . Next, a direct computation reveals that(3.4) Z X g pu k dµ ≤ Z X g pk dµ = 1 (cid:16)R kn λ ( t ) pp − µ − p ( t ) K j ( t )1 − p dt (cid:17) p − < ∞ lassification criteria for regular trees k ≥ n + 1. Since R p = ∞ and µ ( X k ) < ∞ for each k ∈ N , it follows from (3.3)-(3.4)that u k ∈ N ,p ( X ) with u k | X n ≡ k →∞ Z X g pu k dµ = 0 . We thus get Cap p ( X n ) = 0. Conversely, suppose that Cap p ( X n ) = 0. Then there existsa sequence { u k } ∞ k =1 in N ,p ( X ) with u k | X n ≡ k →∞ Z X g pu k dµ = 0 . According to | ( u k − x ) − ( u k − y ) | = | u k ( x ) − u k ( y ) | ≤ Z [ x,y ] g u k ds for p -a.e curve [ x, y ], we obtain that g u k is a p -weak upper gradient of v k := ( u k −
1) andhence g v k ≤ g u k a.e.. Combining this with (3.5) and u k (0) = 1 yields k u k − k p ˙ N ,p ( X ) = k v k k p ˙ N ,p ( X ) = Z X g pv k dµ ≤ Z X g pu k dµ → , as k → ∞ . Therefore u k → N ,p ( X ) with u k ∈ N ,p ( X ), and hence 1 ∈ ˙ N ,p ( X ). Recall that R p = ∞ is equivalent to 1 ∈ ˙ N ,p ( X ) by [21, Theorem 1.3 and Corollary 4.2]. Thus R p = ∞ which completes the proof. Lemma 3.3.
Let X be p -parabolic. Then every nonnegative p -superharmonic function u on X is constant.Proof. Let u ∈ N ,p loc ( X ) be an arbitrary nonnegative p -superharmonic function on X . Weclaim that u is constant. Indeed, let n ∈ N be arbitrary. We denote M := k u k L ∞ ( X n ) . Then
M < ∞ , since u ∈ N ,p loc ( X ) is bounded on X n for each n ∈ N , see the end of Section2.2. By Lemma 3.1, we have Cap p ( X n ) = 0, and hence that there is a sequence { n } ∞ n =1 in N ,p ( X ) with 1 n | X n ≡ n →∞ Z X g p n dµ = 0 . Without loss of generality we assume that spt(1 n ) is compact satisfying (3.6) because1 n ∈ N ,p ( X ). We define a sequence { ϕ n } ∞ n =1 by setting ϕ n ( x ) = max { M · n ( x ) , u ( x ) } − u ( x )6 Khanh Ngoc Nguyen for each n ∈ N and for all x ∈ X . Then(3.7) spt( ϕ n ) ⊂ spt(1 n )for all n ∈ N . We have that 0 ≤ ϕ n ∈ N ,p ( X ) with compact support spt( ϕ n ), because(3.7) and spt(1 n ) is compact. Since u is p -superharmonic on X , it follows that(3.8) Z spt( ϕ n ) g pu dµ ≤ Z spt( ϕ n ) g pu + ϕ n dµ for all n ∈ N . As u + ϕ n = max { M · n , u } , we have that(3.9) g pu + ϕ n ( x ) = g pM · n ( x ) χ { x ∈ X : M · n ≥ u } ( x ) + g pu ( x ) χ { x ∈ X : u>M · n } ( x )for all x ∈ X . According to M = k u k L ∞ ( X n ) , 1 n | X n ≡
1, it follows that u ( x ) ≤ M · n ( x )and M · n ( x ) ≡ M for all x ∈ X n . Thanks to (3.7), we have that for all x ∈ spt(1 n ),(3.10) χ { x ∈ X : u>M · n } ( x ) ≤ χ { x ∈ spt(1 n ) \ X n } ( x ) and g M · n = M · g n χ { x ∈ spt(1 n ) \ X n } . Substituting (3.10) into (3.9) and combining with χ { x ∈ X : M · n ≥ u } ≤ g pu + ϕ n ( x ) ≤ M p · g p n ( x ) χ { x ∈ spt(1 n ) \ X n } ( x ) + g pu ( x ) χ { x ∈ spt(1 n ) \ X n } ( x )for all x ∈ spt(1 n ). By (3.7), the above inequality holds for all x ∈ spt( ϕ n ). Then (3.8)gives that(3.11) Z spt( ϕ n ) g pu dµ ≤ M p Z spt( ϕ n ) \ X n g p n dµ + Z spt( ϕ n ) \ X n g pu dµ for all n ∈ N . By R spt( ϕ n ) \ X n g pu dµ < ∞ , because u ∈ N ,p loc ( X ) and spt( ϕ n ) is compact,subtracting R spt( ϕ n ) \ X n g pu dµ from both sides of (3.11) yields Z X n g pu dµ ≤ M p Z spt( ϕ n ) \ X n g p n dµ ≤ M p Z X g p n dµ for all n ∈ N . Letting n → ∞ , we conclude from (3.6) that R X n g pu dµ = 0 . Since n isarbitrary, this implies that u is constant, and the claim follows. Remark 3.4.
Assume that f > is a p -superharmonic function on X . Then (3.12) Z X f − p g pf ϕ p dµ ≤ (cid:18) pp − (cid:19) p Z X g pϕ dµ for all ϕ ∈ N ,p ( X ) with ≤ ϕ ≤ . This inequality (3.12) is often called a Caccioppoli-type inequality. lassification criteria for regular trees u be an arbitrary nonnegative p -superharmonic function on X . Suppose that X is p -parabolic. By Lemma 3.1, we have that Cap p ( X n ) = 0 for all n ∈ N . Let n ∈ N be arbitrary. Then for any ε > u n,ε ∈ N ,p ( X ) with 0 ≤ u n,ε ≤ u n,ε | X n ≡ Z X g pu n,ε dµ ≤ Cap p ( X n ) + ε = ε. Applying the Caccioppoli inequality 3.12 for f = u + 1 with ϕ = u n,ε , yields(3.14) Z X ( u + 1) − p g pu +1 u pn,ε dµ ≤ (cid:18) pp − (cid:19) p Z X g pu n,ε dµ. Note that g log( u +1) = ( u + 1) − g u +1 by [3, Theorem 2.16 or Proposition 2.17]. We combinethis and (3.13)-(3.14) with u n,ε | X n ≡ Z X n g log( u +1) p dµ ≤ (cid:18) pp − (cid:19) p ε. Letting ε →
0, this gives g log( u +1) = 0 on X n and hence that u is constant on X n . Thus u is constant on X since n ∈ N is arbitrary. The proof of Lemma 3.3 is complete. Proof of Remark 3.4.
Let us first assume that f ≥ ( p − /p . Let ϕ ∈ N ,p ( X ) bearbitrary with 0 ≤ ϕ ≤
1. Then there exists a sequence ϕ n ∈ N ,p ( X ) with 0 ≤ ϕ n ≤ ϕ n ) ⊂ X such that ϕ n → ϕ in N ,p ( X ) as n → ∞ . Hence wemay assume from this and Corollary 2.2 that spt( ϕ n ) ⊂ X n and ϕ n → ϕ a.e. Let w n = f + ϕ pn f − p and g n = (1 − ( p − ϕ pn f − p ) g f + pϕ p − n f − p g ϕ n . By [11, Proposition 6.3.3] or [3, Proof of Proposition 8.8], we obtain that g n is a p -weakupper gradient of w n . Note that 0 ≤ ( p − ϕ pn f − p ≤ f ≥ ( p − /p and 0 ≤ ϕ n ≤
1. We then have by convexity of the function t t p that on Ω = { x ∈ X : ϕ n ( x ) > } , g pn = (cid:18) (1 − ( p − ϕ pn f − p ) g f + ( p − ϕ pn f − p pϕ p − n f − p g ϕ n ( p − ϕ pn f − p (cid:19) p ≤ (1 − ( p − ϕ pn f − p ) g pf + ( p − ϕ pn f − p (cid:18) pϕ p − n f − p g ϕ n ( p − ϕ pn f − p (cid:19) p = (1 − ( p − ϕ pn f − p ) g pf + ( p − − p p p g pϕ n . (3.15)8 Khanh Ngoc Nguyen
Note that 0 ≤ w n − f = ϕ pn f − p ≤ ( p − − pp , because 0 ≤ ϕ ≤ f ≥ ( p − /p ,and spt( w n − f ) ⊂ spt( ϕ n ) ⊂ X n . Then w n − f ∈ L p ( X ) because µ ( X n ) < ∞ . As ϕ n ∈ N ,p ( X ), f ∈ N ,p loc ( X ), Ω ⊂ spt( ϕ n ), spt( ϕ n ) is compact, the estimate (3.15)gives that g f , g n ∈ L p (Ω). Hence we have from g w n − f ≤ g w n + g f ≤ g n + g f on Ω that g w n − f ∈ L p (Ω). Consequently, 0 ≤ w n − f ∈ N ,p (Ω) with compact support spt( w n − f ).Since f is p -superharmonic on Ω, this yields Z Ω g pf dµ ≤ Z Ω g pf + w n − f dµ = Z Ω g pw n dµ. Since g n is a p -weak upper gradient of w n , it follows from (3.15) that Z Ω g pf dµ ≤ Z Ω g pf dµ − ( p − Z Ω ϕ pn f − p g pf dµ + ( p − − p p p Z Ω g pϕ n dµ. Recall that R Ω g pf dµ < ∞ because f ∈ N ,p loc ( X ), Ω ⊂ spt( ϕ n ), and spt( ϕ n ) is compact.Subtracting R Ω g pf dµ < ∞ from both sides of the above estimate, yields Z Ω f − p g pf ϕ pn dµ ≤ ( p − − p p p Z Ω g pϕ n dµ. As ϕ n | X \ Ω ≡
0, we have that R X \ Ω f − p g pf ϕ pn dµ = R X \ Ω g pϕ n dµ = 0, and hence adding thisto both sides of the above inequality to obtain that Z X f − p g pf ϕ pn dµ ≤ ( p − − p p p Z X g pϕ n dµ. Letting n → ∞ , by dominated convergence theorem, via ϕ n → ϕ in N ,p ( X ) and almosteverywhere as n → ∞ , the inequality (3.12) follows under the assumption that f ≥ ( p − /p .In the general case, let ε > h = ( p − /p ( f + ε ) /ε where f > p -superharmonic function on X . Then h ≥ ( p − /p is a p -superharmonic function on X andhence the inequality (3.12) holds for p -superharmonic function h . Using the homogeneityof (3.12), we obtain that (3.12) holds for f + ε and letting ε → n ∈ N , we denote E n := (cid:8) x ∈ X : d (0 , x ) ≤ /n (cid:9) , F n := X \ X n . We define the p -capacity of the pair ( E n , F n ), denoted Cap p ( E n , F n ), by setting(3.16) Cap p ( E n , F n ) = inf (cid:26)Z X g pu dµ : u ∈ N ,p ( X ) , u | E n ≡ , u | F n ≡ , ≤ u ≤ (cid:27) . The following lemma follows straightforwardly from the definitions (1.1),(3.16) of p -capacity. lassification criteria for regular trees Lemma 3.5.
Let X be p -hyperbolic. Then the sequence { Cap p ( E n , F n ) } ∞ n =2 is non-increasing and bounded. Moreover, ∞ > Cap p ( X , F ) ≥ Cap p ( E n , F n ) ≥ Cap p ( { } ) > for all n ≥ .Proof. It is clear from (1.1) and (3.16) that { Cap p ( E n , F n ) } ∞ n =2 is non-increasing andCap p ( E n , F n ) ≥ Cap p ( { } ) for all n ≥
2. Hence for all n ≥ p ( E , F ) ≥ Cap p ( E n , F n ) ≥ Cap p ( { } ) . By Lemma 3.1, we obtain that Cap p ( { } ) >
0. We then have by Cap p ( X , F ) ≥ Cap p ( E , F ) that Cap p ( X , F ) ≥ Cap p ( E n , F n ) ≥ Cap p ( { } ) > n ≥
2. To prove the lemma, it suffices to show that(3.17) Cap p ( X , F ) < ∞ . Note that R λ pp − ( t ) µ − p ( t ) K j ( t )1 − p dt > f by setting f | X ≡ , f | F ≡ f ( x ) = 1 − R | x | λ pp − ( t ) µ − p ( t ) K j ( t )1 − p dt R λ pp − ( t ) µ − p ( t ) K j ( t )1 − p dt for all x ∈ X \ X . Then g ( x ) = λ p − ( x ) µ − p ( x ) K j ( x )1 − p R λ pp − ( t ) µ − p ( t ) K j ( t )1 − p dt χ X \ X ( x )is an upper gradient of f , and f is admissible for computing the capacity Cap p ( X , F ).Thus Cap p ( X , F ) ≤ Z X g pf dµ ≤ Z X g p dµ = 1 (cid:16)R λ pp − ( t ) µ − p ( t ) K j ( t )1 − p dt (cid:17) p − < ∞ . This completes the proof.
Lemma 3.6.
Let X be p -hyperbolic, and suppose that ( X n , d, µ ) is doubling and supportsa p -Poincar´e inequality for each n ∈ N . Then there exists a sequence { u n } ∞ n =1 in N ,p ( X ) with u n | E n ≡ , u n | F n ≡ , ≤ u n ≤ such that u n is a nonconstant p -harmonic functionon X n and Z X n g pu n dµ = Cap p ( E n , F n ) . Khanh Ngoc Nguyen
Proof.
Let n ∈ N . By the definition (3.16) of Cap p ( E n , F n ), there exists a sequence { u n,m } ∞ m =1 in N ,p ( X ) with u n,m | E n ≡ u n,m | F n ≡
0, 0 ≤ u n,m ≤ p ( E n , F n ) ≤ Z X g pu n,m dµ ≤ Cap p ( E n , F n ) + 1 m . By Lemma 3.5, we have(3.19) 0 < Cap p ( { } ) ≤ Cap p ( E n , F n ) < Cap p ( X , F ) < ∞ for any n ≥ { g u n,m } ∞ m =1 is bounded in L p ( X ). We have from µ ( X n +1 ) < ∞ ,0 ≤ u n,m ≤ { u n,m } ∞ m =1 is bounded in L p ( X n +1 ). Then { u n,m } ∞ m =1 is bounded in N ,p ( X n +1 ). We note that N ,p ( X n +1 ) is a reflexive space since ( X n +1 , d, µ ) is doublingand supports a p -Poincar´e inequality, see [5, Theorem 4.48]. Hence Theorem 2.6 givesthat there is a subsequence { u n,m k } ∞ k =1 which converges weakly to some u n ∈ N ,p ( X n +1 ) . By Mazur’s Lemma 2.7, there is a sequence of convex combinations f k which convergesto u n in N ,p ( X n +1 ):(3.20) f k := N k X i = k a i,k u n,i where a i,k ≥ , P N k i = k a i,k = 1, u n,i ∈ { u n,m k } ∞ k =1 . We may assume that f k ( x ) convergespointwise to u n ( x ) as k → ∞ on X n +1 . It is easy to see that u n | E n ≡ u n | X n +1 ∩ F n ≡ u n,i | E n ≡ u n,i | X n +1 ∩ F n ≡ i ≥ k . Now, we extend u n by zerooutside X n +1 and then(3.21) u n ∈ L p ( X ) with u n | E n ≡ , u n | F n ≡ , ≤ u n ≤ . Next, we have by the convexity of the function t t p that Z X n g pf k dµ ≤ N k X i = k a i,k Z X n g pu n,i dµ. As this and the triangle inequality, we have that (cid:18)Z X n g pu n dµ (cid:19) p ≤ (cid:18)Z X n g pf k − u n dµ (cid:19) p + (cid:18)Z X n g pf k dµ (cid:19) p ≤ (cid:18)Z X n g pf k − u n dµ (cid:19) p + N k X i = k a i,k Z X n g pu n,i dµ ! p . (3.22)According to (3.18) and P N k i = k a i,k = 1 yields N k X i = k a i,k Z X n g pu n,i dµ ! p ≤ N k X i = k a i,k Cap p ( E n , F n ) + N k X i = k a i,k i ! p lassification criteria for regular trees ≤ (cid:18) Cap p ( E n , F n ) + 1 k (cid:19) p . (3.23)Substituting (3.23) into (3.22) and combining with f k → u n in N ,p ( X n ) as k → ∞ , weobtain that (cid:18)Z X n g pu n dµ (cid:19) p ≤ (cid:18)Z X n g pf k − u n dµ (cid:19) p + (cid:18) Cap p ( E n , F n ) + 1 k (cid:19) p → Cap p ( E n , F n ) p as k → ∞ . Then the above estimate gives from (3.19) and u n | F n ≡ u n | X \ X n ≡ g u n ∈ L p ( X ). Combining this with (3.21) yields u n ∈ N ,p ( X ) with u n | E n ≡ u n | F n ≡ ≤ u n ≤ u n is admissible for computing the capacity Cap p ( E n , F n ). Itfollows from this and the above estimate that(3.24) Z X n g pu n dµ = Cap p ( E n , F n )for all n ≥
2. We conclude from (3.19),(3.21),(3.24) that there is a nonconstant function u n ∈ N ,p ( X ) with u n | E n ≡ , u n | F n ≡
0, and 0 ≤ u n ≤ < Z X n g pu n dµ = Cap p ( E n , F n ) < ∞ . Finally, we only need to show that u n is a p -harmonic function on X n . Let ϕ bean arbitrary element of N ,p ( X n ) with compact support spt( ϕ ) ⊂ X n . By choosing v = max { , min { , u n + ϕ }} we have that v ∈ N ,p ( X n ) with v | E n ≡ , v | F n ≡ , ≤ v ≤ ϕ ) ⊂ X n and (3.21). It follows from the definition (3.16) of Cap p ( E n , F n )that Cap p ( E n , F n ) ≤ Z X n g pv dµ ≤ Z X n g pu n + ϕ dµ. Combining this with (3.24), we obtain that Z X n g pu n dµ ≤ Z X n g pu n + ϕ dµ for all ϕ ∈ N ,p ( X n ) with compact support spt( ϕ ) ⊂ X n . Hence u n is p -harmonic on X n ,and the claim follows. Lemma 3.7.
Let X be p -hyperbolic, and suppose that ( X n , d, µ ) is doubling and supportsa p -Poincar´e inequality for each n ∈ N . Then there exists a nonconstant nonnegativebounded p -harmonic function u on X with < R X g pu dµ < ∞ .Proof. We first prove that there exists a nonnegative bounded p -harmonic function u on X . To do this, Lemma 3.5 and Lemma 3.6 give that there exists a sequence { u n } ∞ n =2 in2 Khanh Ngoc Nguyen N ,p ( X ) with u n | E n ≡ u n | F n ≡
0, 0 ≤ u n ≤ u n is a nonconstant p -harmonicfunction on X n and(3.25) 0 < Cap p ( { } ) ≤ Z X n g pu n dµ = Cap p ( E n , F n ) ≤ Cap p ( X , F ) < ∞ for all n ≥
2. Let n be arbitrary. It follows from the local H¨older continuity of p -harmonicfunctions (see Theorem 2.11) that { u n } n ≥ n is equibounded and locally equicontinuous on X n . By the Arzel`a-Ascoli theorem, there exists a subsequence, still denoted { u n } n ≥ n ,that converges to u locally uniformly in X n as n → ∞ . Since n is arbitrary, by unique-ness of locally uniform convergence, we may assume that u n converges to u locally uni-formly in X as n → ∞ . Then { u i } i ≥ n and {− u i } i ≥ n are sequences of p -superharmonicfunctions on X n which converge locally uniformly to u and − u in X n for each n re-spectively, and hence by Theorem 2.10 we have that u and − u are p -superharmonicfunctions on X n for all n . We then obtain from Theorem 2.9 that u and − u are p -superharmonic functions on X . Thus u is a nonnegative bounded p -harmonic function on X since 0 ≤ u n ≤ < Z X g pu dµ < ∞ . It follows from (3.25) that { g u n } ∞ n =2 is a bounded sequence in the reflexive space L p ( X ),and hence Theorem 2.6 and Mazur’s Lemma 2.7 give that there exists g ∈ L p ( X ) anda convex combination sequence ¯ g n = P N n i = n a i,n g u i with a i,n ≥ , P N n i = n a i,n = 1 such that¯ g n → g in L p ( X ) as n → ∞ . By Fuglede’s Lemma 2.5, we obtain that there is asubsequence, still denoted ¯ g n , such thatlim n →∞ Z [ x,y ] ¯ g n ds = Z [ x,y ] gds for p -a.e curve [ x, y ]. Note that ¯ g n = P N n i = n a i,n g u i is a p -weak upper gradient of ¯ u n = P N n i = n a i,n u i and ¯ u n converges to u locally uniformly in X as n → ∞ . Hence | u ( x ) − u ( y ) | = lim n →∞ | ¯ u n ( x ) − ¯ u n ( y ) | ≤ lim n →∞ Z [ x,y ] ¯ g n ds = Z [ x,y ] gds for p -a.e curve [ x, y ]. Then g is a p -weak upper gradient of u and hence g u ≤ g a.e..Combining this with ¯ g n → g in L p ( X ) as n → ∞ and the convexity of the function t t p yields(3.26) Z X g pu dµ ≤ Z X g p dµ = lim n →∞ Z X ¯ g pn dµ ≤ lim n →∞ N n X i = n a i,n Z X g pu i dµ. lassification criteria for regular trees { Cap p ( E n , F n ) } ∞ n =2 is a nonincreasing sequence by Lemma 3.5. Hence we haveby u i | F i ≡ u i | X \ X i ≡ n →∞ N n X i = n a i,n Z X g pu i dµ = lim n →∞ N n X i = n a i,n Z X i g pu i dµ = lim n →∞ N n X i = n a i,n Cap p ( E i , F i ) ≤ lim n →∞ N n X i = n a i,n Cap p ( E n , F n )= lim n →∞ Cap p ( E n , F n ) < ∞ . Substituting the above estimate into (3.26) yields R X g pu dµ < ∞ .It remains to show that R X g pu dµ >
0. The preceding being understood, we argue bycontradiction and assume that R X g pu dµ = 0, and hence we can suppose that u is constanton X . Note that ¯ g n = P N n i = n a i,n g u i with a i,n ≥ , P N n i = n a i,n = 1 such that ¯ g n → g in L p ( X )as n → ∞ and(3.27) lim n →∞ Z [ x,y ] ¯ g n ds = Z [ x,y ] gds for p -a.e curve [ x, y ]. By Section 2.2, the empty family is the only curve family withzero p -modulus, it follows that (3.27) holds for any curve [ x, y ]. Moreover, ¯ g n is a p -weak upper gradient of ¯ u n = P N n i = n a i,n u i and ¯ u n is admissible for computing the capacityCap p ( E N n , F N n ), and so Z X ¯ g pn dµ ≥ Z X g p ¯ u n dµ ≥ Cap p ( E N n , F N n ) . Combining this with (3.25) and using ¯ g n → g in L p ( X ) as n → ∞ yields(3.28) Z X g p dµ > . Let ε > x, y ] be an arbitrary curve in X . We have from u n converges to u locally uniformly in X that there exist positive constants N, r onlydepending on x, y such that for all n ≥ N sup t ∈ B ( x,r ) | u n ( t ) − u ( t ) | < ε, sup t ∈ B ( y,r ) | u n ( t ) − u ( t ) | < ε where B ( x, r ) , B ( y, r ) are balls with center x, y and radius r respectively. The aboveestimates give from u is constant that for all n ≥ N , | u n ( x ) − u n ( y ) | ≤ | u n ( x ) − u ( x ) | + | u ( y ) − u n ( y ) | < ε. Khanh Ngoc Nguyen
By Section 2.2, u n is absolute continuous on [ x, y ] and g u n ( z ) = | u ′ n ( z ) | /λ ( z ) for z ∈ [ x, y ] where u ′ n ( z ) on [ x, y ] such that | u n ( x ) − u n ( y ) | = Z [ x,y ] | u ′ n ( z ) | dz. Hence the minimal upper gradient g u n of u n satisfies Z [ x,y ] g u n ds ≤ ε for all n ≥ N . Since (3.27) holds for curve [ x, y ], we have that Z [ x,y ] gds = lim n →∞ Z [ x,y ] ¯ g n ds = lim n →∞ N n X i = n a i,n Z [ x,y ] g u i ds < ε. 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 contracts to (3.28). This completes the proof. Proof of Theorem 1.1. X is p -parabolic ⇔ (2 . ) is given by Lemma 3.1.(1 . ) ⇔ (2 . ) ⇔ (3 . ) is given by Lemma 3.2. Proof of Theorem 1.3. X is p -parabolic ⇒ (1 . ) is given by Lemma 3.3.(1 . ) ⇒ (2 . ) is trivial.(2 . ) ⇒ (3 . ): Let u be a bounded p -harmonic function on X . Then there exists aconstant C > u + C is a nonnegative p -harmonic function on X . Hence u + C is constant by the assumption and so u is constant.(3 . ) ⇒ (4 . ): Let u be an arbitrary p -harmonic function on X with R X g pu dµ < ∞ . ByProposition 7.1.18 in [11], we obtain that(3.29) lim n →∞ Z X g pu − u n dµ = 0where u n = max { min { u, n } , − n } . In fact, we have that(3.30) Z X g pu dµ ≤ p Z X g pu n dµ + 2 p Z X g pu − u n dµ for all n . Note that u n are bounded p -harmonic functions on X and hence u n are constantby the assumption. It follows from (3.29)-(3.30) and u n are constant that Z X g pu dµ = 0and hence u is constant.(4 . ) ⇒ (5 . ) is trivial.(5 . ) ⇒ X is p -parabolic is given by Lemma 3.7.(6 . ) ⇔ X is p -parabolic is given by Theorem 2.13. lassification criteria for regular trees Example 3.8.
Let < p < ∞ . There exists a p -hyperbolic K -regular tree X with adistance and a “non-radial” measure such that R p = ∞ . Let us begin with some notation. For simplicity, let X be a dyadic tree (which means K = 2). Then the root 0 of our tree has two closest vertexes, denoted v and v . Wedenote T = [0 , v ] ∪ { x ∈ X : v ∈ [0 , x ] } and T = [0 , v ] ∪ { x ∈ X : v ∈ [0 , x ] } . Note that the union of T and T is our tree. Given λ i , µ i : [0 , ∞ ) → (0 , ∞ ) with λ i , µ i ∈ L ([0 , ∞ )), for i = 1 ,
2. We introduce a measure µ and a metric d via ds bysetting dµ ( x ) = µ i ( | x | ) d | x | , ds ( x ) = λ i ( | x | ) d | x | , for all x ∈ T i , for i = 1 ,
2. To obtain what we desire, we choose λ ≡ µ ≡ λ ≡ , µ ( x ) = 2 − j ( x ) . Define a metric d and a measure µ as above. Then R p | T := 12 Z ∞ λ pp − ( t ) µ − p ( t )2 j ( t )1 − p dt < ∞ ,R p | T := 12 Z ∞ λ pp − ( t ) µ − p ( t )2 j ( t )1 − p dt = ∞ , and R p = R p | T + R p | T = ∞ . By Theorem 1.1 for the subtree T with R p | T < ∞ ,we obtain that T is p -hyperbolic. Hence there exists a compact set O in T such thatCap T p ( O ) > T p ( O ) := inf (cid:26)Z T g pu dµ : u | O ≡ , u ∈ N ,p ( T ) (cid:27) . Let O be a compact set in T such that Cap T p ( O ) >
0. Then O is bounded in T by Corollary 2.2. Let u ∈ N ,p ( X ) be an arbitrary function with u | O ≡
1. It followsfrom u ∈ N ,p ( X ) that there exists a sequence u n ∈ N ,p ( X ) with compact supportspt( u n ) ⊂ X such that u n → u in N ,p ( X ) as n → ∞ . By Corollary 2.2, we may assumethat spt( u n ) ⊂ X n for each n . Hence spt( u n ) ∩ T ⊂ X n ∩ T , and so spt( u n ) ∩ T is compactin T because spt( u n ) ∩ T is a closed set in T and X n ∩ T is compact in T . Then for each n , u n ∈ N ,p ( T ) with compact support spt( u n ) ∩ T ⊂ T such that u n → u in N ,p ( T )as n → ∞ , and hence u ∈ N ,p ( T ) with u | O ≡
1. We then have by u ∈ N ,p ( T ) with u | O ≡ T p ( O ) that Z X g pu dµ ≥ Z T g pu dµ ≥ Cap T p ( O ) . Khanh Ngoc Nguyen
Since u ∈ N ,p ( X ) with u | O ≡ p ( O ) ≥ Cap T p ( O ) . Combining this with Cap T p ( O ) >
0, we have Cap p ( O ) >
0. Thus X is p -hyperbolic. Example 3.9.
Let < p < ∞ . There exist both p -hyperbolic and p -parabolic K -regulartrees ( X, d, µ ) that are doubling and support a p -Poincar´e inequality. We begin with the p -hyperbolic case. Let µ ( t ) = e − βj ( t ) and λ ( t ) = e − εj ( t ) with ε, β > K < β < log K + εp . It is obvious that µ ( X ) < ∞ and R p < ∞ . More precisely,since log K < β < log K + εp we have that µ ( X ) = Z ∞ µ ( t ) K j ( t ) dt = Z ∞ e − ( β − log K ) j ( t ) dt < ∞ and R p = Z ∞ λ ( t ) pp − µ ( t ) − p K j ( t )1 − p dt = Z ∞ e ( β − log K − εp ) j ( t ) p − dt < ∞ . As R p < ∞ , by Theorem 1.1, it follows that ( X, d, µ ) is a hyperbolic metric measurespace. By [4, Section 3 and Section 4] or [22, Section 2] for log
K < β , we obtain that(
X, d, µ ) supports a doubling measure and a (1 , X, d, µ )is doubling and supports a p -Poincar´e inequality.For the p -parabolic case, let µ ( t ) = e − βj ( t ) and λ ( t ) = e − εj ( t ) with ε, β > β = log K + εp . It is easy to see that ( X, d, µ ) is a doubling p -parabolic K -regular treethat supports a p -Poincar´e inequality by a similar argument as above. Acknowledgement
The author thanks his advisor Professor Pekka Koskela for helpful discussions. The authoralso would like to thank Zhuang Wang for reading the manuscript and giving commentsthat helped to improve the paper.
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