Dyadic Norm Besov-Type Spaces as Trace Spaces on Regular Trees

In this paper, we study function spaces defined via dyadic energies on the boundaries of regular trees. We show that correct choices of dyadic energies result in Besov-type spaces that are trace spaces of (weighted) first order Sobolev spaces.


Introduction
Over the past two decades, analysis on general metric measure spaces has attracted a lot of attention, e.g., [2,4,12,13,[15][16][17]. Especially, the case of a regular tree and its Cantor-type boundary has been studied in [3]. Furthermore, Sobolev spaces, Besov spaces and Triebel-Lizorkin spaces on metric measure spaces have been studied in [5,25,26] via hyperbolic fillings. A related approach was used in [23], where the trace results of Sobolev spaces and of related fractional smoothness function spaces were recovered by using a dyadic norm and the Whitney extension operator.
Dyadic energy has also been used to study the regularity and modulus of continuity of space-filling curves. One of the motivations for this paper is the approach in [20]. Given a continuous g : S 1 → R n , consider the dyadic energy (1.1) Here, {I i,j : i ∈ N, j = 1, · · · , 2 i } is a dyadic decomposition of S 1 such that for every fixed i ∈ N, {I i,j : j = 1, · · · , 2 i } is a family of arcs of length 2π/2 i with j I i,j = S 1 . The next generation is constructed in such a way that for each j ∈ {1, · · · , 2 i+1 }, there exists a unique number k ∈ {1, · · · , 2 i }, satisfying I i+1,j ⊂ I i,k . We denote this parent of I i+1,j by I i+1,j and set I 1,j = S 1 for j = 1, 2. By g A , A ⊂ S 1 , we denote the mean value g A = − A g dH 1 = 1 H 1 (A) A g dH 1 . One could expect to be able to use the energy Eq. 1.1 to characterize the trace spaces of some Sobolev spaces (with suitable weights) on the unit disk. On the contrary, the results in [23] suggest that the trace spaces of Sobolev spaces (with suitable weights) on the unit disk should be characterized by the energy (1.2) where I i,0 = I i,2 i , and the example g(x) = χ I 1,1 shows that E(g; p, λ) is not comparable to E(g; p, λ). Notice that the energies (1.1) and (1.2) can be viewed as dyadic energies on the boundary of a binary tree (2-regular tree). More precisely, for a 2-regular tree X in Section 2.1 with = log 2 in the metric (2.1), the measure ν on the boundary ∂X is the Hausdorff 1-measure by Proposition 2.10. Furthermore, there is a one-to-one map h from the dyadic decomposition of S 1 to the dyadic decomposition of ∂X defined in Section 2.4, which preserves the parent relation, i.e., h( I ) = h(I ) for all dyadic intervals I of S 1 . Since every point in S 1 is the limit of a sequence of dyadic intervals, we can define a maph from S 1 to ∂X by mapping any point x = k∈N I k in S 1 to the limit of {h(I k )} k∈N (if the limit is not unique for different choices of sequence {I k } for x, then just pick one of them). It follows from the definition of ∂X that the maph is an injective map. Since the measure ν is the Hausdorff 1measure and ∂X \h(S 1 ) is a set of countably many points, it follows from the definition of Hausdorff measure that ν(∂X \h(S 1 )) = 0. Since diam(I ) ≈ diam(h(I )) for any dyadic interval I of S 1 and we can use dyadic intervals to cover a given set in the definition of a Hausdorff measure, there is a constant C ≥ 1 such that for any measurable set A ⊂ S 1 . Then one could expect to be able to use an energy similar to Eq. 1.2, theḂ 1/p,λ p -energy given by to characterize the trace spaces of suitable Sobolev spaces of the 2-regular tree. This turns out to hold in the sense that any function in L p (∂X) with finiteḂ 1/p,λ p -energy can be extended to a function in a certain Sobolev class.
However, there exists a Sobolev function whose trace function has infiniteḂ 1/p,λ p -energy. More precisely, let 0 be the root of the tree X and let x 1 , x 2 be the two children of 0. We define a function u on X by setting u(x) = 0 if the geodesic from 0 to x passes through x 1 , u(x) = 1 if the geodesic from 0 to x passes through x 2 and define u to be linear on the geodesic [x 1 , x 2 ] = [0, x 1 ] ∪ [0, x 2 ]. Then u is a Sobolev function on X with the trace function g = χ h(I 1,1 ) whoseḂ 1/p,λ p -energy is not finite for any λ ≥ −1, since the energy (1.2) of the function χ I 1,1 is not finite for any λ ≥ −1. But the energy (1.1) of the function χ I 1,1 is finite. Hence, rather than studying the energy (1.3), we shall work with an energy similar to Eq. 1.1. We define the dyadicḂ 1/p,λ p energy by setting where Q = ∪ j ∈N Q j is a dyadic decomposition on the boundary of the 2-regular tree in Section 2.4. Instead of only considering the above dyadic energy on the boundary of a 2-regular tree, we introduce a general dyadic energyḂ θ,λ p in Definition 2.12, defined on the boundary of any regular tree and for any 0 ≤ θ < 1. It is natural to ask whether the Besov-type space B θ,λ p (∂X) in Definition 2.12 defined via theḂ θ,λ p -energy is a trace space of a suitable Sobolev space defined on the regular tree. We refer to [1,9,10,14,18,19,23,24,[27][28][29][30] for trace results on Euclidean spaces and to [3,21,25] for trace results on metric measure spaces.
In [3], the trace spaces of the Newtonian spaces N 1,p (X) on regular trees were shown to be Besov spaces defined via double integrals. Our first result is the following generalization of this theorem.
The measure μ λ above is defined in Eq. 2.2 by dμ λ (x) = e −β|x| (|x| + C) λ d|x|, and the space N 1,p (X, μ λ ) is a Newtonian space defined in Section 2.3. If λ = 0, then N 1,p (X, μ λ ) = N 1,p (X) and Theorem 1.1 recovers the trace results from [3] for the Newtonian spaces N 1,p (X). Here and throughout this paper, for given Banach spaces X(∂X) and Y(X), we say that the space X(∂X) is a trace space of Y(X) if and only if there is a bounded linear operator T : Y(X) → X(∂X) and there exists a bounded linear extension operator E : X(∂X) → Y(X) that acts as a right inverse of T , i.e., T • E = Id on the space X(∂X).
We required in Theorem 1.1 that p > (β − log K)/ > 0. The assumption that β − log K > 0 is necessary in the sense that we need to make sure that the measure μ λ on X is doubling; see Section 2.2. The requirement that p > (β − log K)/ will ensure that θ > 0. So it is natural to consider the case p = (β − log K)/ ≥ 1.
Moreover, for any p = (β − log K)/ ≥ 1, there exists a bounded nonlinear extension operator E : L p (∂X) → N 1,p (X) so that the trace operator T defined via limits of E(f ) along geodesic rays for f ∈ L p (∂X) satisfies T • E = Id on L p (∂X).
A result similar to Theorem 1.2 for the weighted Newtonian space N 1,p ( , ω dμ) with a suitable weight ω has been established in [21] provided that is a bounded domain that admits a p-Poincaré inequality and whose boundary ∂ is endowed with a p-codimensional Ahlfors regular measure. In Theorem 1.2, for the case p = (β − log K)/ > 1, we require that λ > p − 1 to ensure the existence of limits along geodesic rays. In the case p = (β − log K)/ = 1, these limits exist even for λ = 0, and there is a nonlinear extension operator that acts as a right inverse of the trace operator, similarly to the case of W 1,1 in Euclidean setting; see [10,24].
However, except for the case p = 1 and λ = 0, Theorem 1.2 does not even tell whether the trace operator T is surjective or not: N 1,p (X, μ λ ) is a strict subset of N 1,p (X) when λ > 0. In the case p = (β−log K)/ = 1 and λ > 0, the trace operator T is actually not surjective, and we can find a Besov-type space B α 1 (∂X) (see Definition 2.14) which is the trace space of the Newtonian space N 1,1 (X, μ λ ). We stress that B α 1 (∂X) and B 0,λ 1 (∂X) are different spaces. More precisely, B 0,λ 1 (∂X) is a strict subspace of B α 1 (∂X), see Proposition 3.8 and Example 3.9.
Trace results similar to Theorem 1.3 in the Euclidean setting can be found in [11,30]. The second part of Theorem 1.2 asserts the existence of a bounded nonlinear extension operator from L p (∂X) to N 1,p (X) whenever p = (β − log K)/ ≥ 1. Nonlinearity is natural here since results due to Peetre [24] (also see [8]) indicate that, for p = 1 and λ = 0, one can not find a bounded linear extension operator that acts as a right inverse of the trace operator in Theorem 1.2. On the other hand, the recent work [22] gives the existence of a bounded linear extension operator E from a certain Besov-type space to BV or to N 1,1 such that T •E is the identity operator on this Besov-type space, under the assumption that the domain satisfies the co-dimension 1 Ahlfors-regularity. The extension operator in [22] is a version of the Whitney extension operator. This motivates us to further analyze the operator E from Theorem 1.1: it is also of Whitney type. The co-dimension 1 Ahlfors-regularity does not hold for our regular tree (X, μ λ ), but we are still able to establish the following result for N 1,p (X, μ λ ) with p ≥ 1 for our fixed extension operator E. Moreover, the space B 0,λ p (∂X) is the optimal space for which E is both bounded and linear, i.e., if X ⊂ L 1 loc (∂X) is a Banach space so that the extension operator E : X → N 1,p (X, μ λ ) is bounded and linear and so that T • E is the identity operator on X, then X is a subspace of B 0,λ p (∂X).
The optimality of the space B 0,λ p (∂X) is for the explicit extension operator E in Theorem 1.4. The space B 0,λ p (∂X) may not be the optimal space unless we consider this particular extension operator. For example, for p = 1 and λ > 0, the optimal space is B α 1 (∂X) rather than B 0,λ 1 by Theorem 1.3. This splitting happens since the two extension operators from Theorems 1.3 and 1.4 are very different: the latter one is of Whitney type in the sense that the extension to an edge is based on the average of the boundary function over the dyadic "shadow" of size comparable to that of the edge, while the former one uses the average over a dyadic boundary element for the definition of the extension to several edges of different sizes.
The paper is organized as follows. In Section 2, we give all the preliminaries for the proofs. More precisely, we introduce regular trees in Section 2.1 and we consider the doubling condition on a regular tree X and the Hausdorff dimension of its boundary ∂X. We introduce the Newtonian spaces on X and the Besov-type spaces on ∂X in Sections 2.3 and 2.4, respectively. In Section 3, we give the proofs of all the above mentioned theorems, one by one.
In what follows, the letter C denotes a constant that may change at different occurrences.

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 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 X a rooted tree if it has a distinguished vertex called the root, which we will denote by 0. The neighbors of a vertex x ∈ X 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 is a rooted tree such that each vertex has exactly K children. Then all 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 regular if it is a K-ary tree for some K ≥ 1.
For x ∈ X, let |x| be the distance from the root 0 to x, that is, the length of the geodesic from 0 to x, where the length of every edge is 1 and we consider each edge to be an isometric copy of the unit interval. The geodesic connecting two vertices x, y ∈ V is denoted by [x, y], and its length is denoted |x − y|. If |x| < |y| and x lies on the geodesic connecting 0 to y, we write x < y and call the vertex y a descendant of the vertex x. More generally, we write x ≤ y if the geodesic from 0 to y passes through x, and in this case |x −y| = |y|−|x|.
Let > 0 be fixed. We introduce a uniformizing metric (in the sense of Bonk-Heinonen-Koskela [6], see also [3]) on X by setting Here d |z| 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. In this metric, diamX = 2/ if X is a K-ary tree with K ≥ 2.
Next we construct the boundary of the regular K-ary tree by following the arguments in [3,Section 5]. We define the boundary of a tree X, denoted ∂X, by completing X with respect to the metric d X . An equivalent construction of ∂X is as follows. An element ξ in ∂X is identified with an infinite geodesic in X starting at the root 0. Then we may denote ξ = 0x 1 x 2 · · · , where x i is a vertex in X with |x i | = i, and x i+1 is a child of x i . Given two points ξ, ζ ∈ ∂X, there is an infinite geodesic [ξ, ζ ] connecting ξ and ζ . Then the distance of ξ and ζ is the length (with respect to the metric d X ) of the infinite geodesic [ξ, ζ ]. More precisely, if ξ = 0x 1 x 2 · · · and ζ = 0y 1 y 2 · · · , let k be an integer with x k = y k and x k+1 = y k+1 . Then by Eq. 2.1 The restriction of d X to ∂X is called the visual metric on ∂X in Bridson-Haefliger [7]. The metric d X is thus defined onX. 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 ω. Moreover, balls in X will be denoted B(x, r), while B(ξ, r) stands for a ball in ∂X.
Throughout the paper we assume that 1 ≤ p < +∞ and that X is a K-ary tree with K ≥ 2 and metric d X defined as in Eq. 2.1.

Doubling Condition on X and Hausdorff Dimension of ∂X
The first aim of this section is to show that the weighted measure Here the lower bound of the constant C will make the estimates below simpler. If λ = 0, then dμ 0 (x) = e −β|x| d|x| = dμ(x), which coincides with the measure used in [3]. If β ≤ log K, then μ λ (X) = ∞ for the regular K-ary tree X by Eq. 2.4 below. Hence X would not be doubling as X is bounded.
Next we estimate the measures of balls in X and show that our measure is doubling. Let denote an open ball in X with respect to the metric d X . Also let F (x, r) = {y ∈ X : y ≥ x and d X (x, y) < r} denote the downward directed "half ball". The following algebraic lemma and the relation between a ball and a "half ball" come from [3, Lemma 3.1 and 3.2].

Lemma 2.2
For every x ∈ X and r > 0 we have where z ≤ x and In the above lemma, z is the largest (in the ≤ relationship) common ancestor of B(x, r), i.e., we have z ≤ y for any y ∈ B(x, r).
We begin to estimate the measure of the ball B(x, r) and of the half ball F (z, r).
Note that for each |z| ≤ t ≤ |z| + ρ, the number of points y ∈ F (z, r) with |y| = t is approximately K t−|z| . Hence , Hence we obtain that It is easy to check that for any ρ > 0 and z ∈ X, we have that Therefore, Then for any z ∈ X and ρ > 0, Hence we obtain that Proof For any x ∈ X and 0 < r ≤ e − |x| / , let z be as in Lemma 2.2. If z = 0, then Moreover, in both cases, since r < e − |x| / , by Lemma 2.2, we have Combing Eq. 2.7 with the fact that in both cases 1 ≤ e |x|−|z| ≤ (1 + re |x| ) 1/ ≈ 1, the result follows by applying Lemma 2.3 to F (x, r) and F (z, 2r) (or F (0, r + ρ) for z = 0).
where [z, x] denotes the geodesic in the tree X joining x and z.
Hence it remains to show that For any z ∈ X and x ∈ X with z ≤ x, we have that where |x| = ∞ if x ∈ ∂X. Then by using an argument similar to the estimate in Lemma 2.3, which implies that for any t ≥ 0, Hence we obtain that Comparing the estimate (2.8) with the estimate (2.5), since ρ = |x| − |z|, e ρ log K ≥ 1 and which induces that

Corollary 2.6 Let x ∈ X and z be as in Lemma 2.2. Then if
Then Lemma 2.3 implies Towards the another direction, by Eq. 2.3 and Lemma 2.5, we have that Moreover, we have where t = re |x| . Here in the last inequality we used the fact that re |x| ≥ 1. Hence we obtain that μ λ (B(x, r)) e −β|z| (|z| + C) λ . Combing the above inequality with Eq. 2.10, we finish the proof of Since re |x| ≥ 1, we know that It then follows from Eq. 2.3 that Hence we obtain that e −β|z| (|z| + C) λ ≈ r β/ (|z| + C) λ , which finishes the proof.  (B(x, r)) ≈ r.
In the case |x| ≥ (log 2)/ with 2r ≥ R 0 , if r ≥ R 0 , then Let z r and z 2r be defined as in Lemma 2.2 with respect to r and 2r. From Corollary 2.4 and the above estimates, the doubling condition of μ λ follows once we prove that |z r | + C |z 2r | + C ≈ 1. (2.11) which gives Eq. 2.11. If 2r ≤ (1 − e − |x| )/ , for C ≥ 2(log 2)/ , we obtain that which gives that |z r | + C ≤ 2(|z 2r | + C). Combining with the fact that |z 2r | ≤ |z r |, Eq. 2.11 is obtained. Therefore we finish the proof of this corollary.
The following result is given by [3, Lemma 5.2].

Proposition 2.10
The boundary ∂X is an Ahlfors Q-regular space with Hausdorff dimension Hence we have an Ahlfors Q-regular measure ν on ∂X with ν(B(ξ, r)) ≈ r Q = r log K/ , for any ξ ∈ ∂X and 0 < r ≤ diam∂X.

Newtonian Spaces on X
Let u ∈ L 1 loc (X, μ λ ). We say that a Borel function g : whenever z, y ∈ X and γ is the geodesic from z to y, where ds X denotes the arc length measure with respect to the metric d X . In the setting of a tree any rectifiable curve with end points z and y contains the geodesic connecting z and y, and therefore the upper gradient defined above is equivalent to the definition which requires that inequality (2.12) holds for all rectifiable curves with end points z and y.
The notion of upper gradients is due to Heinonen and Koskela [16]; we refer interested readers to [12,17] for a more detailed discussion on upper gradients.
The Newtonian space N 1,p (X, μ λ ), 1 ≤ p < ∞, is defined as the collection of all the functions for which where the infimum is taken over all upper gradients of u.

Besov-Type Spaces on ∂X via Dyadic Norms
We first recall the Besov space B θ p,p (∂X) defined in [3].
Next, we give a dyadic decomposition on the boundary ∂X of the K-ary tree X: Let V n = {x n j : j = 1, 2, · · · , K n } be the set of all n-level vertices of the tree X for any n ∈ N, where a vertex x is n-level if |x| = n. Then we have that We denote by Q the set {I x : x ∈ V } and Q n the set {I x : x ∈ V n } for any n ∈ N. Then Q 0 = {∂X} and we have Q = n∈N Q n .
Then the set Q is a dyadic decomposition of ∂X. Moreover, for any n ∈ N and I ∈ Q n , there is a unique element I in Q n−1 such that I is a subset of it. It is easy to see that if I = I x for some x ∈ V n , then I = I y with y the unique parent of x in the tree X. Hence the structure of the tree X gives a corresponding structure of the dyadic decomposition of ∂X which we defined above.
Since we want to characterize the trace spaces of the Newtonian spaces with respect to our measure μ λ , we introduce the following Besov-type spaces B θ,λ p (∂X). For λ > 0, we next define special Besov-type spaces with θ = 0 and p = 1. Before the definition, we first fix a sequence {α(n) : n ∈ N} such that there exist constants c 1 ≥ c 0 > 1 satisfying (2.13) A simple example of such a sequence is obtained by letting α(n) = 2 n . Definition 2.14 For λ > 0, the Besov-type space B α 1 (∂X) consists of all functions f ∈ L 1 (∂X) for which theḂ α 1 -dyadic energy of f defined as Here for any I = I x ∈ Q α(n) with x ∈ V α(n) and n ≥ 1, we denote I = I y where y ∈ V α(n−1) is the ancestor of x in X. The norm on B α

Remark 2.15
Actually, the choice of the sequence {α(n)} n∈N will not affect the definition of B α 1 (∂X): by Theorem 1.3 we obtain that any two choices of the sequences {α(n)} n∈N lead to comparable norms, for more details see Corollary 3.7.

Proof of Theorem 1.1
Proof Trace Part: Let f ∈ N 1,p (X, μ λ ). We first define the trace operator as where the limit is taken along the geodesic ray [0, ξ). Then our task is to show that the above limit exists for ν-a.e. ξ ∈ ∂X and that the trace Tr f satisfies the norm estimates. Let ξ ∈ ∂X be arbitrary and let x j = x j (ξ ) be the ancestor of ξ with |x j | = j . To show that the limit in Eq. 3.1 exists for ν-a.e. ξ ∈ ∂X, it suffices to show that the functioñ is the geodesic ray from 0 to ξ and g f is an upper gradient of f .
To be more precise, iff * ∈ L p (∂X), we have |f * | < ∞ for ν-a.e. ξ ∈ ∂X, and hence the limit in Eq. 3.1 exists for ν-a.e. ξ ∈ ∂X. Set r j = 2e −j / . Then on the edge [x j , x j +1 ] we have the relations where the comparison constants depend on , β. Then we obtain the estimatẽ For p = 1, the above estimates are also true without using the Hölder inequality. It follows that for p ≥ 1, Integrating over all ξ ∈ ∂X, since ν(∂X) ≈ 1, we obtain by means of Fubini's theorem that Notice that χ [x j (ξ ),x j +1 (ξ )] (x) is nonzero only if j ≤ |x| ≤ j + 1 and x < ξ. Thus the last estimate can be rewritten as where E(x) = {ξ ∈ ∂X : x < ξ} and j (x) is the largest integer such that j (x) ≤ |x|.

It follows from [3, Lemma 5.1] that E(x) = B(ξ, r) for any ξ ∈ E(x)
and r ≈ e − j (x) . Hence we obtain from Proposition 2.10 that ν(E(x)) ≈ r Q j (x) . Since p(1 − κ) > β/ − log K/ = β/ − Q, then for any j (x) ∈ N, we have that Hence we obtain thatf * is in L p (∂X), which gives the existence of the limit in Eq. 3.1 for ν-a.e. ξ ∈ ∂X. In particular, since |f | ≤f * , we have the estimate and hence the norm estimate To estimate the dyadic energy f pḂ θ,λ p (∂X) , for any I ∈ Q n , ξ ∈ I and ζ ∈ I , we have where x j = x j (ξ ) and y j = y j (ζ ) are the ancestors of ξ and ζ with |x j | = |y j | = j , respectively. In the above inequality, we used the fact that x n−1 (ξ ) = y n−1 (η). By using Eq. 3.3 and an argument similar to Eq. 3.4, we obtain that Choose 0 < κ < θ and insert r κ j r −κ j into the above sum. If p > 1, then the Hölder inequality and Eq. 3.3 imply that For p = 1 the estimates above is also true without using the Hölder inequality. It follows from Fubini's theorem and from ν(I ) ≈ ν( I ) that Using the notation E(x) and j (x) defined before, the above estimate can be rewritten as Since e − n ≈ r n−1 and p − β/ + Q = θp, we obtain the estimate Here the last inequality employed the estimate j n=0 r κp−θp n (n + 1) λ r κp−θp j which comes from the facts r n = 2e − n / and κp − θp < 0. Thus, we obtain the estimate which together with Eq. 3.5 finishes the proof of Trace Part. Extension Part: Let u ∈ B θ,λ p (∂X). For x ∈ X with |x| = n ∈ N, let where I x ∈ Q n is the set of all the points ξ ∈ ∂X such that the geodesic [0, ξ) passes through x, that is, I x consists of all the points in ∂X that have x as an ancestor. By Eqs. 3.1 and 3.6 we notice that Trũ(ξ ) = u(ξ ) whenever ξ ∈ ∂X is a Lebesgue point of u. If y is a child of x, then |y| = n + 1 and I x is the parent of I y . We extendũ to the edge [x, y] as follows: For each t ∈ [x, y], set andũ (t) =ũ(x) + gũ(t)d X (x, t). (3.8) Then we define the extension of u to beũ.
Since gũ is a constant andũ is linear with respect to the metric d X on the edge [x, y], it follows that |gũ| is an upper gradient ofũ on the edge [x, y]. We have that [x,y] |gũ| p dμ λ ≈ n+1 n |u I y − u I y | p e −βτ + np (τ + C) λ dτ ≈ e (−β+ p)(n+1) (n + 1) λ |u I y − u I y | p . (3.9) Now sum up the above integrals over all the edges on X to obtain that . (3.10) To estimate the L p -norm ofũ, we first observe that for any t ∈ [x, y]. Then we obtain the estimate Since β − Q = β − log K > 0, the sum of e −βn+ nQ n λ converges. Hence we obtain the L p -estimate X |ũ| p dμ λ ∂X |u| p dν.
Proof By using the estimate (3.11), for x, y ∈ X with y a child of x and |x| = j , we obtain that Since u is Lipschitz on ∂X, then for any ξ, ζ ∈ ∂X, Hence, for any I ∈ Q j , we have that It follows that The construction of the extension operator is given by gluing the N 1,p extensions in Lemma 3.4 of Lipschitz approximations of the boundary data with respect to a sequence of layers on the tree X. The main idea of the construction is inspired by [21,Section 7] and [22,Section 4] whose core ideas can be traced back to Gagliardo [10] who discussed extending functions in L 1 (R n ) to W 1,1 (R n+1 + ).
Proof of Proposition 3.5 Let f ∈ L p (∂X). We approximate f in L p (∂X) by a sequence of (∂X) . Note that this requirement of rate of convergence of f k to f ensures that f k → f pointwise ν-a.e. in ∂X. For technical reasons, we choose f 1 ≡ 0.
Then we choose a decreasing sequence of real numbers {ρ k } +∞ k=1 such that • ρ k ∈ {e − n / : n ∈ N}; • 0 < ρ k+1 ≤ ρ k /2; • k ρ k LIP (f k , ∂X) ≤ C f L p (∂X) . These will now be used to define layers in X. Let ψ k (x) = max 0, min 1, We denote − log( ρ k )/ by [ρ k ]. This is a integer satisfying e − [ρ k ] / = ρ k . Then we obtain 0 ≤ ψ k ≤ 1 and that (3.14) For any Lipschitz function f k , we can define the extensionf k of f k by using Eqs. 3.6, 3.7 and 3.8. Then we define the extension of f as It follows from Eq. 3.14 that for any In order to obtain the L p -estimate of an upper gradient off , it suffices to consider the L p -estimate of Lipf , where for any function u, Lip u(x) is defined as We first apply the product rule for locally Lipschitz function, which yields that Thus,

It follows from Lemma 3.3 that
Recall thatũ is affine one any edge of X, with "slope" gũ, for the extensionũ given via Eqs. 3.6, 3.7 and 3.8, for any function u. Hence Lipũ = gũ. Therefore, it follows from Lemma 3.4 that Here in the last inequality, we used the defining properties of {ρ k } +∞ k=1 . Thus, we have shown that Altogether, we obtain that where α −1 (j ) is the largest integer m such that α(m) ≤ j . Since λ > 0 and we obtain the estimate Hence we obtain the estimate Thus, we obtain the norm estimate which finishes the proof of the Trace Part. Extension Part: Let u ∈ B α 1 (∂X). Since α(0) is not necessarily zero, we let α(−1) = 0. For any x ∈ X with |x| = α(n) and −1 ≤ n ∈ Z, let where I x ∈ Q is the set of all the points ξ ∈ ∂X such that the geodesic [0, ξ) passes through x, that is, I x consists of all the points in ∂X that have x as an ancestor.
Next, we compare the function spaces B α 1 (∂X) and B 0,λ 1 (∂X). Proof Let f ∈ L 1 (∂X). For any I ∈ Q α(n) with n ∈ R, define the set J I := {I ∈ Q : I ⊂ I I }.
Since the sum of 1/i λ+1 converges for λ > 0, f is well defined for all ξ ∈ ∂X and is bounded. Moreover, for any vertex x = 0τ 1 · · · τ k , it follows from the definition of f that Therefore, for the vertex x above, we have Hence theḂ 0,λ 1 -energy of f is On the other hand, for any I ∈ Q α(n) , we have where τ i ∈ {0, 1} depends on I . We define a random series X α(n) by setting where (σ i ) i are independent random variables with common distribution P (σ i = 1) = P (σ i = −1) = 1/2. Since the measure ν is a probability measure which is uniformly distributed on ∂X, it follows from Eq. 3.22 that I ∈Q α(n) ν(I )|f I − f I | = E(|X α(n) |).
Here E(|X α(n) |) is the expected value of |X α(n) |. By the Cauchy-Schwarz inequality, E(|X α(n) |) ≤ (E(X 2 α(n) )) 1/2 , we have that Here the second to last equality holds since σ i and σ j are independent for i = j and E(σ i σ j ) = E(σ i )E(σ j ) = 0 for i = j . Define α(n) = 2 n . Then we obtain that