Trace Operators on Regular Trees

exists for almost every ξ ∈ ∂Bn(0, 1). Here almost everywhere refers to the surface measure on ∂Bn(0, 1). In this sense, u has a well de ned trace almost everywhere on ∂Bn(0, 1). Towards a more constructive de nition of a trace, let us extend u to a function Eu ∈ W1,1(Rn). This is possible by classical extension theorems in [5, 24]. By the version of Lebesgue di erentiation theorem for Sobolev functions [26, Section 5.14], the limit


Introduction
Let us begin with the classical setting. Consider the unit ball B n ( , ) in the n-dimensional Euclidean space R n . If u belongs to the usual Sobolev space W , (B n ( , )) consisting of all integrable functions whose all rst order distributional derivatives are also integrable over B n ( , ), then u has a representative v for which the limit lim t→ v(tξ ) (1.1) exists for almost every ξ ∈ ∂B n ( , ). Here almost everywhere refers to the surface measure on ∂B n ( , ). In this sense, u has a well de ned trace almost everywhere on ∂B n ( , ). Towards a more constructive de nition of a trace, let us extend u to a function Eu ∈ W , (R n ). This is possible by classical extension theorems in [5,24]. By the version of Lebesgue di erentiation theorem for Sobolev functions [26, Section 5.14], the limit lim r→ mn(B(x, r))ˆB (x,r) Eu dmn exists for H n− -almost every x. Here mn is the Lebesgue measure on R n and H n− refers to the (n − )dimensional Hausdor measure. It then follows from the (1,1)-Poincaré inequality that also lim r→ mn(B(x, r) ∩ B n ( , ))ˆB (x,r)∩B n ( , ) u dmn (1.2) exists for H n− -almost every x and also that, for almost every ξ ∈ ∂B n ( , ) there is a value Tu(ξ ) for which lim r→ mn(B(ξ , r) ∩ B n ( , ))ˆB (ξ ,r)∩B n ( , ) |u(x) − Tu(ξ )| dmn(x) = . (1.3) Thus we have three di erent possible traces, but it turns out that Tu(ξ ) coincides with the limits in (1.1) and (1.2) (for a suitable v) almost everywhere on ∂B n ( , ). Moreover, by the (q, p)-Poincaré inequality (with Notice that Γx is also a K-regular tree if x is a vertex, obviously with root x.

De nition 1.2. Let ≤ q < ∞ and G be a K-regular tree with metric d λ and measure µ as above, with µ(G) < ∞. Fix a function f de ned on G. We say that the Lebesgue-point-type trace T L f of f on ∂G exists if T L f (ξ ) := lim
x ξ →ξ µ(Γx ξ )ˆΓ x ξ f (y) dµ(y) (1.5) exists for ν-a.e ξ ∈ ∂G. We say that the boundary trace of f of order q on ∂G exists if there is a function Tq f : ∂G → R so that for ν-a.e ξ ∈ ∂G.
One can nd versions of the two notions of traces in De nition 1.2 in literature under various names. We refer the readers to [7,Chapter 2], [19,Section 6.6], [20,Section 9.6], [26,Section 3.1] for discussions in the setting of Euclidean spaces, and [16][17][18] (also the references therein) for discussions in the setting of metric measure spaces. Notice that in the setting of a Muckenhoupt Ap-weight discussed above, the analogs of the traces T R f , T L f and Tq f , ≤ q ≤ p, exist and actually coincide with each other almost everywhere on ∂B n ( , ).
It is then natural to ask whether T R f , T L f , Tq f exist (for suitable q) and coincide for a given function f ∈ N ,p (G, d λ , µ). Towards this, we recall a concept introduced in [14]. Let ≤ p < ∞. We set and where j(t) is the largest integer such that j(t) ≤ |x| + . Since we work with a xed pair λ, w, we will usually refer to Rp(λ, w) simply by Rp . One should view Rp as an analog of the isoperimetric pro le of a Riemannian manifold in [11][12][13]. We assume in what follows that λ p w − ∈ L /(p− ) loc ([ , ∞)) to make sure that the niteness of Rp is a condition at in nity.
Our rst result shows that the existence of any of T R f , T L f , Tq f , ≤ q ≤ p, for all f ∈ N ,p (G) is equivalent to the niteness of Rp . Moreover, all these di erent traces of f coincide when Rp < ∞. Theorem 1.3. Let ≤ p < ∞ and G be a K-regular tree with metric d λ and measure µ as above. Assume µ(G) < ∞ and let ≤ q ≤ p. Then the following are equivalent: Moreover, if one of T R f , T L f , Tq f exists for each f ∈ N ,p (G), then all of them exist and coincide ν-a.e on ∂G for a given f .
As a direct consequence of Theorem 1.3 we see that the existence of the trace operator Tq is independent of the value of q ∈ [ , p]. We do not know if one could even obtain this for all q ∈ [ , p + ϵ] for some ϵ > only depending on p, Rp(λ, w), λ, w.
Based on the discussion in the beginning of our introduction, one should nd Theorem 1.3 somewhat surprising since it does not seem possible to extend our functions to a larger underlying nice space and the niteness of Rp should not, in general, imply the validity of Poincaré inequalities. In fact, the validity of Poincaré inequalities under a doubling condition on (G, d λ , µ) has very recently been characterized via a Muckenhoupt-type condition in [22]. The reason why we do not need a Poincaré inequality or a doubling measure and do not need to move to a representative when we consider T R is basically that our space is locally one-dimensional. Our second result deals with the coincidence of N ,p (G) andṄ ,p (G). HereṄ ,p (G) is the homogeneous version of N ,p (G). Theorem 1.4. Let ≤ p < ∞ and G be a K-regular tree with metric d λ and measure µ with µ(G) < ∞ as above. Suppose that Rp < ∞. Then N ,p (G) =Ṅ ,p (G).
Consequently, Theorem 1.3 could alternatively be stated forṄ ,p (G). In the case where µ(G) = ∞, the homogeneous version of our Sobolev space is much larger than the non-homogeneous one. However, even under the assumption that µ(G) < ∞, Rp < ∞ is not a necessary condition for N ,p (G) =Ṅ ,p (G). Example 3.8 in Section 3 shows that there exists a K-regular tree (G, d λ , µ) so that Rp = ∞ and µ(G) < ∞ but nevertheless N ,p (G) =Ṅ ,p (G).
The paper is organized as follows. In Section 2, we introduce K-regular trees and their boundaries, and Newtonian spaces. In Section 3, we give the proofs of Theorem 1.3 and Theorem 1.4.
Throughout this paper, the letter C (sometimes with a subscript) will denote positive constants that usually depend only on our space and may change at di erent occurrences; if C depends on a, b, . . . we write For any function f ∈ L loc (G) and any measurable subset A ⊂ G of positive measure, we let ffl A fdµ stand for µ(A)´A fdµ.

Preliminaries . Regular trees and their boundaries
A graph G is a pair (V , E), where V is a set of vertices and E is a set of edges. We call a pair of vertices x, y ∈ V neighbors if x is connected to y by 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 G is a connected graph without cycles.
We call a tree G a rooted tree if it has a distinguished vertex called the root, which we will denote by . The neighbors of a vertex x ∈ V are of two types: the neighbors that are 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. Then all vertices except the root of G have degree K + , and the root has degree K. We say that a tree G is K-regular if it is a K-ary tree for some K ≥ .
Let G be a K-regular tree with a set of vertices V and a set of edges E for some K ≥ . For simplicity of notation, we let X = V ∪ E and call it a K-regular tree. The geodesic connecting x, y ∈ X is denoted by [x, y].
For any x, y ∈ X, let |x − y| be the metric graph distance from x to y, that is, the metric graph length of the geodesic [x, y] given by We denote by |x| the metric graph distance from the root to x. Then the metric graph distance between two vertices is the number of edges needed to connect them. Given a curve γ, we say that γ is an in nite geodesic in X if γ is a simple curve and l G (γ) = ∞.
On our K-regular tree X, we de ne a measure µ and a metric d λ by setting where λ, w : [ , ∞) → ( , ∞) are xed with λ, w ∈ L loc ([ , ∞)). For any two points x, y ∈ X, the distance between x and y, denoted d λ (x, y), is where [x, y] is the unique geodesic between x, y. In particular, if x ∈ [ , y] then the distance between [x, y] is given by The measure of our K-regular tree is where j(t) is the largest integer such that j(t) ≤ t + .
We abuse notation and let w(x) and λ(x) denote w(|x|) and λ(|x|), respectively, for any x ∈ X, if there is no danger of confusion. We refer the interested readers to [14, 21, Section 2] for a discussion on this metric and this measure.
A tree is the quintessential Gromov hyperbolic space, and hence we can consider the visual boundary of the tree as in Bridson-Hae iger [3]. We de ne the boundary of our K-regular tree X, denoted ∂X, as the collection of all in nite geodesics in X starting at the root . Given two points ξ , ζ ∈ ∂X, there is an in nite geodesic (ξ , ζ ) in X connecting ξ and ζ .
To avoid confusion, points in X are denoted by Latin letters such as x, y and z, while for points in ∂X we use Greek letters such as ξ , ζ and η.
Given z ∈ X, we de ne the subtree with respect to the root z, denoted Γz, by setting Let ∂Γz be the collection of ξ ∈ ∂X with respect to all the in nite geodesics (in X) containing z and starting at the root . Then ∂Γz := {ξ ∈ ∂X : z ∈ [ , ξ )}.
We equip ∂X with the natural probability measure ν as in Falconer [6] by distributing the unit mass uniformly on ∂X. Then for any subset A ⊂ ∂X, the boundary measure of A, denoted by ν(A), is For any x ∈ X with |x| = j, if we denote by Ix (or ∂Γx) the set {ξ ∈ ∂X : the geodesic [ , ξ ) passes through x}, then ν(Ix) = ν(∂Γx) = K −j . We refer to [2, Lemma 5.2] for more information on our boundary measure ν.
Recall that a metric space (∂X, d b ) is an ultrametric space if for each triple of points ξ , ζ , η ∈ ∂X we have Proposition 2.1. The metric space (∂X, d b ) is an ultrametric space under the assumption that´∞ λ(t) dt < ∞ and hence any two closed balls in ∂X are either disjoint or contain one another.
Proof. For any ξ , ξ , ξ ∈ ∂X, we let x (ξ i ,ξ j ) be the last common point of [ , ξ i ) and [ , ξ j ) for each i, j ∈ { , , }. Let k i,j = |x (ξ i ,ξ j ) | for each i, j ∈ { , , }. Then k ≥ min{k , k } and for any triple of points ξ , ξ , ξ ∈ ∂X. Thus (∂X, d b ) is an ultrametric space. The latter part of the proposition is a direct consequence of the ultrametric property of ∂X. The proof is complete.
By Proposition 2.1, any two closed balls in ∂X are either disjoint or contain one another. Then (X, d b , ν) is a Vitali metric measure space, i.e every subset A of ∂X and for every covering B of A by closed balls satisfying By the Lebesgue di erentiation theorem on a Vitali metric measure space in [10, Section 3.4], we obtain the following theorem.

. Newtonian spaces
Let ≤ p < ∞ and X be a K-regular tree with metric d λ and measure µ as in Section 2.1. Let f ∈ L loc (X, d λ , µ). We say that a Borel function g : whenever y, z ∈ X and γ is the geodesic from y to z. In the setting of our tree, any recti able curve with end points z and y contains the geodesic connecting z and y, and therefore the upper gradient de ned above is equivalent to the de nition which requires that (2.1) holds for all recti able curves with end points z and y. The notion of upper gradients was introduced in [9]. We refer the interested readers to [1,8,10,23] for a more detailed discussion on upper gradients.
The Newtonian space N ,p (X) := N ,p (X, d λ , µ), ≤ p < ∞, is de ned as the collection of all the functions f with nite N ,p -norm where the in mum is taken over all upper gradients of u. If f ∈ N ,p (X), then it is continuous by (2.1); recall here our standing assumption that λ p w − ∈ L /(p− ) loc ([ , ∞)). We de ne the homogeneous Newtonian spacesṄ ,p (X), ≤ p < ∞, as the collection of all the continuous functions f that have an upper gradient ≤ g ∈ L p (X). The homogeneousṄ ,p -norm is given by Here is the root of our K-regular tree X and the in mum is taken over all upper gradients of f .

Proofs of Theorem 1.3-1.4
In this section, if we do not speci cally mention, we always assume that ≤ p < ∞ and that X is a K-regular tree with metric d λ and measure µ as in Section 2.1, with µ(X) < ∞.
Let us rst prove that Rp(λ, w) < ∞ together with µ(X) < ∞ guarantee that our metric space is bounded.
Proof. For p > , the Hölder inequality giveŝ Notice that´∞ w(t)K j(t) dt is precisely µ(X) and that the second term is R p− p p . Hence the claim follows for p > since µ(X) < ∞ and Rp < ∞. For p = , a similar idea gives´∞ λ(t)dt ≤ µ(X)R < ∞. The proof is complete.
In what follows, the notation x ξ means that x ξ ∈ [ , ξ ). We set Lemma 3.1 in [14], applied to the subtree Γz where z ∈ X, gives the following identity.

Lemma 3.2.
Let u ∈ L p (X). For any z ∈ X, we have that We also need the following formulation of Theorem 1.1 in [14]. We begin by establishing the existence of two of the asserted limits. Proof. Suppose that µ(X) < ∞ and Rp < ∞. Let f ∈ N ,p (X) and g f ∈ L p (X) be an upper gradient of f . By Lemma 3.3, we obtain that T R f exists. To prove that Tq f exists, it su ces to show that lim holds for ν-a.e ξ ∈ ∂X. By the Hölder inequality and the dominated convergence theorem, it follows from ≤ q ≤ p, (1.4), and (2.1) that for any x ξ ∈ [ , ξ ), To obtain (3.1), we only need to show that lim Suppose rst that p > . By the Hölder inequality, a direct computation reveals that for any [x, y] in X, ˆ[ and For p = , by an argument similar to (3.3), without using the Hölder inequality, we also obtain that for any [x, y] in X,ˆ[ (3.6) and hence that (3.4) and (3.5) also hold for p = .
Applying Lemma 3.2 for Γz = X and u = g f , it follows from g f ∈ L p (X) that for ν-a.e ξ ∈ ∂X. We conclude from (3.5) and (3.7) that lim for ν-a.e ξ ∈ ∂X. In order to get (3.2), we next estimate H (x ξ ). By the Fubini theorem, (3.4) gives that andˆ∞ g(t)λ(t)dt = ∞. (3.15) Pick n so thatˆn g(t)λ(t)dt = . (3.16) As µ(X \ X n ) = lim l →∞ µ((X \ X n ) ∩ X l ), we nd l ∈ N with n ≤ l such that Since µ(X \ X n ) = K n µ(Γ xn(ξ ) ) and µ(( for any ξ ∈ ∂X and for any n, m ∈ N with n ≤ m, the above estimates give for any ξ ∈ ∂X. By (3.15) we nd m with l ≤ m such that Hence we have by (3.17) that for any ξ ∈ ∂X. We continue by choosing n with k ≤ n such that n k g(t)λ(t)dt = . (3.21) By induction on n , l , m , k , n with n ≤ l ≤ m ≤ k ≤ n , there exist four sequences  , we have that f is continuous, ≤ f ≤ , and g is an upper gradient of f . By (3.14) and the fact that µ(X) < ∞, it follows that f ∈ N ,p (X). Combining (3.18), (3.20), (3.22),(3.23), we conclude that for any ξ ∈ ∂X, for any i ∈ N, Thus (3.13) holds. The claim follows.
It is obvious that µ(X) < ∞ and Rp = ∞ for any ≤ p < ∞. Indeed, since (β − log K) > εp > we have that For any ≤ p < ∞, as (β − K − εp) > we obtain that As in the proof of Theorem 1.4 we have that N ,p (X) ⊂Ṅ ,p (X). Hence we only need to prove thatṄ ,p (X) ⊂ N ,p (X). It su ces to show that for any f ∈Ṅ ,p (X), Let g f be an upper gradient of f . For p > , we have by the Hölder inequality that for any x ∈ X, where = β − log K .