Admissibility versus $A_p$-conditions on regular trees

We show that the combination of doubling and $(1,p)$-Poincare inequality is equivalent to a version of the $A_p$-condition on rooted K-ary trees.


Introduction
The class of p-admissible weights for Sobolev spaces and di erential equations on R n was introduced in [12]. The de nition was initially based on four conditions, but Theorem 2 in [10] and Theorem 5.2 in [13] reduce them to the following two conditions, see also [12, 2nd ed., Section 20].
De nition 1.1. A measure µ on R n is p-admissible, ≤ p < ∞, if it is doubling and supports a ( , p)-Poincaré inequality. If dµ = w dx, we also say that the weight w is p-admissible.
Here µ supports a (q, p)-Poincaré inequality, ≤ q < ∞, ≤ p < ∞, if there is a constant C > such that for every u ∈ C (R n ), every x ∈ R n and all r > .
In [12,Section 15], it was shown that Muckenhoupt Ap-weights are p-admissible, but the converse is not true in R n , n ≥ , see also [6]. Surprisingly, on the real line R, any p-admissible measure is actually given by an Ap-weight, see [7]. Very recently, it was also shown in [5] that a measure on R is locally p-admissible if and only if it is given by a local Ap-weight. Moreover, on R n , p-admissible measures can be characterized by a stronger version of the Poincaré inequality, the (q, p)-Poincaré inequality with q > p. Under doubling, the ( , p)-Poincaré inequality improves to a (q, p)-Poincaré inequality with q > p by [10] and any measure satisfying (q, p)-Poincaré inequality with q > p is a doubling measure, see [1] and [17].
In the recent years, analysis on regular trees has been under development, see [3,[18][19][20][21]. Given a Kregular tree X (a rooted K-ary tree), K ≥ , we introduce a metric structure on X by considering each edge of X to be an isometric copy of the unit interval. Then the distance between two vertices is the number of edges needed to connect them and there is a unique geodesic that minimizes this number. Let us denote the root by 0. If x is a vertex, we de ne |x| to be the distance between 0 and x. Since each edge is an isometric copy of the unit interval, we may extend this distance naturally to any x belonging to an edge.
Write d|x| for the length element on X and let µ : [ , ∞) → ( , ∞) be a locally integrable function. We abuse notation and refer also to the measure generated via dµ(x) = µ(|x|)d|x| by µ. Further, let λ : [ , ∞) → ( , ∞) be locally integrable and de ne a distance via ds(x) = λ(|x|)d|x| by setting d(z, y) = [z,y] ds(x) whenever z, y ∈ X and [z, y] is the unique geodesic between z and y. 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. Throughout this paper, we assume additionally that the diameter of X is in nity.
Our space (X, d, µ) is a metric measure space and hence one may de ne a Newtonian Sobolev space N ,p (X) := N ,p (X, d, µ) based on upper gradients [14] and [22]. It is then natural to ask if we can characterize the p-admissibility of a given µ, see Section 2.2 for the de nitions. To do so, we introduce the following Apconditions on regular trees.
Before continuing, we rst introduce some notations. For any x ∈ X and r > , we denote byx r the point Hencex r is an ancestor of x and x r is a descendant of x, see Section 2.1 for more relations between points on regular trees. Also let be the downward directed "half ball". It is perhaps worth to mention that the notationsx r and F(x, r) coincide with the notation "z" and F(x, r) in [3, Lemma 3.2], respectively. Given < p < ∞, we set and we de ne where j(w) and j(x) are the smallest integers such that j(w) ≥ |w| and j(x) ≥ |x|, respectively. Notice that Ap(x, r) is independent of the choice of x r among the points y with x ∈ [ , y] and d(y, x) = r.
De nition 1.2. Let ≤ p < ∞ and X be a K-regular tree with distance d and metric µ. We say that µ satis es the Ap-condition if sup Ap(x, r) : x ∈ X, r > < ∞.
We say that µ satis es the Ap-condition far from if If K = and λ ≡ , then the -regular tree (X, d, µ) is isometric to the half line (R + , dx, µ dx) and our Apcondition (1.3) is equivalent to µ being a Muckenhoupt Ap-weight, see [5][6][7]12] for more information about Muckenhoupt Ap-weights. Above, we call (1.4) "Ap-condition far from " since < r ≤ d( , x) is equivalent to d( , x) ≥ r/ > , which means that x has to be "far" away from the root in terms of r.
The main result of this paper is the following characterization of p-admissibility on regular trees. The characterizations for K = and K ≥ are di erent. For K ≥ , a K-regular tree has a kind of symmetry property with respect to the root , since the root has more than one branch. But for K = , the root behaves like an end point.
Readers who are familiar with the results on the real line R may regard our K-regular tree with K ≥ as a generalized model of the real line R. As a byproduct, a slightly modi ed proof of Theorem 1.3 for K ≥ gives a new proof of [7,Theorem 2]. On the other hand, for K = , one may connect the result on -regular trees with the result on bounded intervals (see [5,Theorem 4.6] for bounded intervals). Hence Theorem 1.3 is new and interesting even when K = and λ ≡ , since it gives a full characterization of p-admissibility on the half line R + .
In [5,Example 4.7], one can nd a weight ω on the interval [ , ] which is -admissible but not a Muckenhoupt A 1 -weight on ( , ). By a suitable constant extension of ω on ( , ∞), we obtain a weight ω which is -admissible but not a Muckenhoupt A 1 -weight on R + . As evidence towards Theorem 1.3 for K = , it is easy to check that the extended weight ω on R + satis es the A 1 -condition far from , i.e., condition (1.4) holds. We refer to [5] and [8] for more details.
Let us close this introduction by pointing out that the constant " " in Ap-condition far from (1.4) is not necessary. Actually replacing by any constant ∞ > c > , Theorem 1.3 for K = holds. Here the requirement of c > is sharp in the sense that there exists an example (R+, dx, µ dx) such that (1.4) holds for any positive constant c < replacing , but µ is not even doubling, see Remark 4.5 and Example 4.6.
The paper is organized as follows. In section 2, we introduce regular trees, p-admissibility and Newtonian spaces on our tree. We give the proof of Theorem 1.3 for K ≥ in Section 3 and the proof of Theorem 1.3 for K = is given in Section 4.

Preliminaries
Throughout this paper, the letter C (sometimes with a subscript) will denote positive constants; if C depends on a, b, . . ., we write C = C(a, b, . . .).

. 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 . 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 + , 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 ≥ .
For x ∈ X, let |x| be the distance from the root to x, that is, the length of the geodesic from to x, where the length of every edge is and we consider each edge to be an isometric copy of the unit interval. The geodesic connecting two points 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 to y, we write x < y and call y a descendant of the point x. More generally, we write x ≤ y if the geodesic from to y passes through x, and in this case |x − y| = |y| − |x|.
On our K-regular tree X, we de ne the metric ds and measure dµ by setting Here d |x| is the measure which gives each edge Lebesgue measure , as we consider each edge to be an isometric copy of the unit interval and the vertices are the end points of this interval. Hence for any two points z, y ∈ X, the distance between them is 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.
Throughout the paper, we let denote the (open) ball in X with center x and radius r, and let σB(x, r) = B(x, σr). Also is the downward directed half ball. For any x ∈ X and r > , we denote byx r the point in Hencex r is the ancestor of any point y ∈ B(x, r). Usually, the choice of x r is not unique, but we will not specify it since the results and proofs in this paper are independent of the choice of x r .

. Admissibility
Let u ∈ L loc (X). We say that a Borel function g : whenever z, y ∈ X and γ is the geodesic from z to y. In the setting of a 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 inequality (2.1) holds for all recti able curves with end points z and y. In [9,15], the notion of a p-weak upper gradient is given. A Borel function g : X → [ , ∞] is called a p-weak upper gradient of u if (2.1) holds on p-a.e. curve. Here we say that a property holds for p-a.e. curve if it fails only for a recti able curve family Γ with zero p-modulus, i.e., there is Borel function ≤ ρ ∈ L p (X) such that γ ρ ds = ∞ for every curve γ ∈ Γ. We refer to [9,15] for more information about p-weak upper gradients. The notion of upper gradients is due to Heinonen and Koskela [14]; we refer interested readers to [2,9,15,22] for a more detailed discussion on upper gradients.
The Newtonian space N ,p (X), for ≤ p < ∞, is de ned as the collection of the functions for which the given norm is nite, where the in mum is taken over all p-weak upper gradients g of u.
A measure µ is doubling if there exists a positive constant C d such that for all balls B(x, r) with x ∈ X and r > , where the constant C d is called the doubling constant. (X, d, µ) supports a ( , p)-Poincaré inequality if there exist positive constants C P > and σ ≥ such that for all balls B(x, r) with x ∈ X and r > , every integrable function u on σB(x, r) and all upper gradients g, 3) is equivalent to the ( , p)-Poincaré inequality given in the Introduction. It perhaps worth to point out that, since our K-regular trees are geodesic spaces, if µ is p-admissible, the dilation constant σ in (2.3) can be taken to , see [10] and [11].

Proof of Theorem 1.3 for K ≥
In this section, we give the proof of Theorem 1.3 for K ≥ . To do so, we establish the following lemmas. Lemma 3.1. Let ≤ p < ∞ and X be a K-regular tree with distance d and measure µ where K ≥ . Assume that µ satis es the Ap-condition. Then µ is p-admissible.
Since µ satis es the Ap-condition, < C A < ∞. Case p = : We rst show that µ is a doubling measure. Let x ∈ X and r > be arbitrary. Notice that A 1 (x, r) ≤ C A . Then it follows from (1.2) that (F(x r , r)) . Hence and that µ(F(x r , r)) ≥ µ (B(x, r)).
It follows from estimate (3.1) that , r)) , which proves that µ is a doubling measure with doubling constant C A since r > and the pair (x, r) is arbitrary.
Next we prove that (X, d, µ) supports a ( , )-Poincaré inequality. Consider an arbitrary ball B(x, r) with x ∈ X and r > . By the triangle inequality, we obtain that for the left-hand side of our Poincaré inequality. By the de nition of upper gradients and the Fubini theorem, for any upper gradient gu of u, the right-hand side of ( . ) rewrites as
Since the measure µ satis es the A 1 -condition, A 1 (x r , r) < C A . It follows from (1.2) that Combining with the fact that K j(x r ) ≤ K j(x r ) , we obtain that for any w ∈ B(x, r). Combining gu dµ for all balls B(x, r). Case p > : Let us rst prove that µ is a doubling measure. Let B(x, r) be an arbitrary ball in X. Since µ satis es the Ap-condition, we have Ap(x, r) ≤ C A , and hence

5)
A simple calculation using the Hölder inequality shows that Inserting (3.5) into the above estimate yields Note that µ(F(x, r)) ≤ µ(B(x, r)) and µ(F(x r , r)) ≥ µ (B(x, r)). Then the estimate (3.6) implies that , which gives that µ is a doubling measure with doubling constant C A p+ , since r > and B(x, r) is arbitrary.
Next we show that (X, d, µ) supports a ( , p)-Poincaré inequality. Suppose B(x, r) is an arbitrary ball with center x ∈ X and radius r > . Since the measure µ satis es the Ap-condition, then Ap(x r , r) < C A . It follows from (1.1) that Recall that the left-hand side of our Poincaré inequality can be estimated by (3.3). A simple calculation shows that for any point w ∈ B(x, r). Inserting the estimate (3.8) into (3.3) yields that (F(x r , r)).
Applying the Hölder inequality for the right-hand side of the above inequality, it follows that By using the estimate (3.7), we obtain that Note that F(x r , r)) ⊂ B(x, r) and that B(x, r) ⊂ F(x r , r). Since µ is a doubling measure with doubling constant C A p+ , we have that Inserting the above estimate into the estimate ( . ), we have Thanks to the estimates ( . ) and ( . ), we obtain Proof. Let x ∈ X and r > be arbitrary. Let ε be an arbitrary positive number. Let x ∈ X be a closest vertex of x with |x | > |x|. Then we de ne Since µ is p-admissible, we may assume that µ satis es the doubling condition (2.2) and the ( , p)-Poincaré inequality (2.3). Case p = : Let In order to test the ( , )-Poincaré inequality (2.3), we de ne < m + ε and a = [x,x r ] χ Eε (w) ds(w). Note that Eε is a non-empty set by the de nition of m and that χ Eε (w)ds(w) > .
By the de nition of u, we obtain that gu := χ Eε is an upper gradient of u. Hence the right-hand side of the ( , )-Poincaré inequality (2.3) is Here the second equality holds since χ Eε (w) is non-zero only if w ∈ F(x, r/ ). Note that µ(σB(x, r)) ≥ µ (B(x, r)). Then it follows from the de nition of Eε that gu dµ ≤ C P r µ (B(x, r)) (m + ε)a. Note that u ≡ on E and u ≡ a on E . Hence, at least one of the following holds: (3.14) Since K ≥ , then E and E are not empty. Notice that Kµ(E ) ≥ µ(F(x, r) \ F(x, r/ )). Furthermore, the doubling property of µ gives Then it follows from (3.14) and (3.15) that the left-hand side of the ( , )-Poincaré inequality (2.3) is Combining the estimates (3.12) and (3.16), we obtain that Since a > and µ(F(x r , r)) ≤ µ(B(x, r)) ≤ C d µ(B(x, r)), it follows that < µ(F(x r , r)) r ≤ C d C P K · (m + ε).
Since ε and the pair (x, r) are arbitrary, letting ε → , the A 1 -condition holds. Case p > : We de ne By the de nition of u, we obtain that is an upper gradient of u. Note that u ≡ on E and u ≡ b on E where E and E are de ned as for p = . Therefore, by an argument similar to the one in p = case, the left-hand side of the ( , p)-Poincaré inequality (2.3) can be estimated as For the right-hand side, we have that Since µ(σB(x, r)) ≥ µ(B(x, r)), it follows that Combining (3.18) and (3.19), we obtain that Notice that µ(F(x r , r)) ≤ µ(B(x, r)) ≤ C d µ (B(x, r)). Hence we have Recalling the de nition of b, the above estimate can be rewritten as Since the pair (x, r) is arbitrary, the above estimate implies that µ satis es the Ap-condition.
where C(C A ) is a constant only depending on C A . In this lemma, since µ only satis es the Ap-condition far from , i.e., for all balls B(x, r) with d( , x) ≥ r/ > . To get that µ is a doubling measure, it is su cient to show that ( . ) holds for all balls B(x, r) with d( , x) < r/ . Note that d( , r ) = r ≥ max{ r/ , r/ , r/ }. Applying ( . ) for B( r , r), B( r , r) and B( r , r) in turns, we obtain that µ(B( r , r)) ≤ Cµ(B( r , r)) ≤ C µ(B( r , r)) ≤ C µ(B( r , r/ )).