Admissibility versus Ap-Conditions on Regular Trees

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Admissibility versus Ap-Conditions on Regular Trees © 2020 Khanh Ngoc Nguyen and Zhuang Wang, published by De Gruyter. Published version Nguyen, Khanh Ngoc; Wang, Zhuang Nguyen, K. N., & Wang, Z. (2020). Admissibility versus Ap-Conditions on Regular Trees. Analysis and Geometry in Metric Spaces, 8(1), 92-105. https://doi.org/10.1515/agms-2020-0110 2020 Open Access. ©2020 Khanh Ngoc Nguyen and Zhuang Wang, published by De Gruyter. This work is licensed under the Creative Commons Attribution alone 4.0 License. Anal. Geom. Metr. Spaces 2020; 8: 92–105 Research Article Open Access Khanh Ngoc Nguyen and Zhuang Wang* Admissibility versus Ap-Conditions on Regular Trees https://doi.org/10.1515/agms-2020-0110 Received December 30, 2019; accepted May 20, 2020. Abstract: We show that the combination of doubling and (1,p)-Poincaré inequality is equivalent to a version of the Ap-condition on rooted K-ary trees. Keywords: Ap-condition; doubling measure; Poincaré inequality; regular tree MSC: 30L99, 31C45, 46E35 Dedicated to Professor Pekka Koskela on the occasion of his 59th birthday celebration 1Introduction The class of p-admissible weights for Sobolev spaces and differential equations on Rnwas introduced in [12]. The definition 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]. Definition 1.1. A measure µon Rnis p-admissible,1≤p<∞, if it is doubling and supports a (1,p)-Poincaré inequality. If dµ =w dx, we also say that the weight wis p-admissible. Here µsupports a (q,p)-Poincaré inequality, 1≤q<∞,1≤p<∞, if there is a constant C>0such that   Z– B(x,r) |u−uB(x,r)|qdµ   1/q ≤Cr   Z– B(x,r) |∇u|pdµ   1/p for every u∈C1(Rn), every x∈Rnand all r>0. In [12, Section 15], it was shown that Muckenhoupt Ap-weights are p-admissible, but the converse is not true in Rn,n≥2, 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 Ris locally p-admissible if and only if it is given by a local Ap-weight. Moreover, on Rn,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 (1,p)-Poincaré inequality improves to a (q,p)-Poincaré inequality with q>pby [10] and any measure satisfying (q,p)-Poincaré inequality with q>pis a doubling measure, see [1] and [17]. In the recent years, analysis on regular trees has been under development, see [3, 18–21]. Given a Kregular tree X(a rooted K-ary tree), K≥1, we introduce a metric structure on Xby considering each edge of Xto be an isometric copy of the unit interval. Then the distance between two vertices is the number of edges needed to connect them and there is a unique geodesic that minimizes this number. Let us denote the root Khanh Ngoc Nguyen, University of Jyväskylä, Jyväskylä, Finland, E-mail: [email protected], [email protected] *Corresponding Author: Zhuang Wang, University of Jyväskylä, Jyväskylä, Finland, E-mail: [email protected] Khanh Ngoc Nguyen and Zhuang Wang, Admissibility versus Ap-Conditions on Regular Trees |93 by 0. If xis a vertex, we define |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 xbelonging to an edge. Write d|x|for the length element on Xand let µ: [0,∞)→(0,∞)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: [0,∞)→ (0,∞)be locally integrable and define a distance via ds(x) = λ(|x|)d|x|by setting d(z,y) = R[z,y]ds(x)whenever z,y∈Xand [z,y]is the unique geodesic between zand 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 Xis infinity. Our space (X,d,µ)is a metric measure space and hence one may define a Newtonian Sobolev space N1,p(X) := N1,p(X,d,µ)based on upper gradients [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 definitions. To do so, we introduce the following Apconditions on regular trees. Before continuing, we first introduce some notations. For any x∈Xand r>0, we denote by ¯ xrthe point in [0,x]with d(¯ xr,x) = min{r,d(0,x)}and denote by xra point in Xsuch that x∈[0,xr]with d(xr,x) = r. Hence ¯ xris an ancestor of xand xris a descendant of x, see Section 2.1 for more relations between points on regular trees. Also let F(x,r) = {y∈X:x∈[0,y],d(x,y)<r} be the downward directed “half ball". It is perhaps worth to mention that the notations ¯ xrand F(x,r)coincide with the notation “z" and F(x,r)in [3, Lemma 3.2], respectively. Given 1<p<∞, we set Ap(x,r) = µ(F(¯ xr,2r)) 2r·   1 rZ [x,xr] Kj(w)−j(x)µ(w) λ(w)!1 1−p ds(w)   p−1 (1.1) and we define A1(x,r) = µ(F(¯ xr,2r)) 2r·ess supw∈[x,xr] λ(w) Kj(w)−j(x)µ(w)(1.2) 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 xramong the points ywith x∈[0,y]and d(y,x) = r. Definition 1.2. Let 1≤p<∞and Xbe a K-regular tree with distance dand metric µ. We say that µsatisfies the Ap-condition if sup Ap(x,r) : x∈X,r>0<∞.(1.3) We say that µsatisfies the Ap-condition far from 0if sup Ap(x,r) : x∈X,0<r≤8d(0,x)<∞.(1.4) If K= 1 and λ≡1, then the 1-regular tree (X,d,µ)is isometric to the half line (R+,dx,µ dx)and our Apcondition (1.3) is equivalent to µbeing a Muckenhoupt Ap-weight, see [5–7, 12] for more information about Muckenhoupt Ap-weights. Above, we call (1.4) “Ap-condition far from 0” since 0<r≤8d(0,x)is equivalent to d(0,x)≥r/8>0, which means that xhas to be “far” away from the root 0in terms of r. The main result of this paper is the following characterization of p-admissibility on regular trees. Theorem 1.3. Let 1≤p<∞and Xbe a K-regular tree with distance dand measure µ. Then we have: 1. For K= 1,µis p-admissible if and only if µsatisfies the Ap-condition far from 0. 2. For K≥2,µis p-admissible if and only if µsatisfies the Ap-condition. The characterizations for K= 1 and K≥2are different. For K≥2, a K-regular tree has a kind of symmetry property with respect to the root 0, since the root has more than one branch. But for K= 1, the root 0behaves like an end point. 94 |Khanh Ngoc Nguyen and Zhuang Wang, Admissibility versus Ap-Conditions on Regular Trees Readers who are familiar with the results on the real line Rmay regard our K-regular tree with K≥2as a generalized model of the real line R. As a byproduct, a slightly modified proof of Theorem 1.3 for K≥2gives a new proof of [7, Theorem 2]. On the other hand, for K= 1, one may connect the result on 1-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= 1 and λ≡1, since it gives a full characterization of p-admissibility on the half line R+. In [5, Example 4.7], one can find a weight ωon the interval [0,1] which is 1-admissible but not a Muckenhoupt A1-weight on (0,1). By a suitable constant extension of ωon (1,∞), we obtain a weight ω0which is 1-admissible but not a Muckenhoupt A1-weight on R+. As evidence towards Theorem 1.3 for K= 1, it is easy to check that the extended weight ω0on R+satisfies the A1-condition far from 0, 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 “8” in Ap-condition far from 0(1.4) is not necessary. Actually replacing 8by any constant ∞>c>1, Theorem 1.3 for K= 1 holds. Here the requirement of c>1is sharp in the sense that there exists an example (R+,dx,µ dx)such that (1.4) holds for any positive constant c0<1replacing 8, but µis 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≥2in Section 3 and the proof of Theorem 1.3 for K= 1 is given in Section 4. 2Preliminaries Throughout this paper, the letter C(sometimes with a subscript) will denote positive constants; if Cdepends on a,b,. . ., we write C=C(a,b,. . .). 2.1 Regular trees and their boundaries Agraph Gis a pair (V,E), where Vis a set of vertices and Eis a set of edges. We call a pair of vertices x,y∈V neighbors if xis connected to yby an edge. The degree of a vertex is the number of its neighbors. The graph structure gives rise to a natural connectivity structure. A tree 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 Xarooted tree if it has a distinguished vertex called the root, which we will denote by 0. The neighbors of a vertex x∈Xare of two types: the neighbors that are closer to the root are called parents of xand all other neighbors are called children of x. Each vertex has a unique parent, except for the root itself that has none. AK-ary tree is a rooted tree such that each vertex has exactly Kchildren. 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 0to x, that is, the length of the geodesic from 0to x, where the length of every edge is 1and we consider each edge to be an isometric copy of the unit interval. The geodesic connecting two points x,y∈Vis denoted by [x,y], and its length is denoted |x−y|. If |x|<|y| and xlies on the geodesic connecting 0to y, we write x<yand call ya descendant of the point x. More generally, we write x≤yif the geodesic from 0to ypasses through x, and in this case |x−y|=|y|−|x|. On our K-regular tree X, we define the metric ds and measure dµ by setting dµ =µ(|x|)d|x|,ds(x) = λ(|x|)d|x|, where λ,µ: [0,∞)→(0,∞)with λ,µ∈L1 loc([0,∞)). Here d|x|is the measure which gives each edge Lebesgue measure 1, as we consider each edge to be an isometric copy of the unit interval and the vertices Khanh Ngoc Nguyen and Zhuang Wang, Admissibility versus Ap-Conditions on Regular Trees |95 are the end points of this interval. Hence for any 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 zto yin X. We abuse the notation and let µ(x)and λ(x)denote µ(|x|)and λ(|x|), respectively, for any x∈X, if there is no danger of confusion. Throughout the paper, we let B(x,r) = {y∈X:d(x,y)<r} denote the (open) ball in Xwith center xand radius r, and let σB(x,r) = B(x,σr). Also F(x,r) = {y∈X:x∈[0,y],d(x,y)<r} is the downward directed half ball. For any x∈Xand r>0, we denote by ¯ xrthe point in [0,x]with d(¯ xr,x) = min{r,d(0,x)}and denote by xra point in Xsuch that x∈[0,xr]with d(xr,x) = r. Hence ¯ xris the ancestor of any point y∈B(x,r). Usually, the choice of xris not unique, but we will not specify it since the results and proofs in this paper are independent of the choice of xr. 2.2 Admissibility Let u∈L1 loc(X). We say that a Borel function g:X→[0,∞]is an upper gradient of uif |u(z)−u(y)|≤Z γ g ds (2.1) whenever z,y∈Xand γis the geodesic from zto y. In the setting of a tree any rectifiable curve with end points zand ycontains the geodesic connecting zand y, and therefore the upper gradient defined above is equivalent to the definition which requires that inequality (2.1) holds for all rectifiable curves with end points zand y. In [9, 15], the notion of a p-weak upper gradient is given. A Borel function g:X→[0,∞]is called a p-weak upper gradient of uif (2.1) holds on p-a.e. curve. Here we say that a property holds for p-a.e. curve if it fails only for a rectifiable curve family Γwith zero p-modulus, i.e., there is Borel function 0≤ρ∈Lp(X)such that Rγρ 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 N1,p(X), for 1≤p<∞, is defined as the collection of the functions for which the given norm kukN1,p(X):=  Z X |u|pdµ + inf gZ X |g|pdµ  1/p is finite, where the infimum is taken over all p-weak upper gradients gof u. A measure µis doubling if there exists a positive constant Cdsuch that for all balls B(x,r)with x∈Xand r>0, µ(B(x,2r)) ≤Cdµ(B(x,r)),(2.2) where the constant Cdis called the doubling constant. (X,d,µ)supports a (1,p)-Poincaré inequality if there exist positive constants CP>0and σ≥1such that for all balls B(x,r)with x∈Xand r>0, every integrable function uon σB(x,r)and all upper gradients g, Z– B(x,r) |u−uB(x,r)|dµ ≤CPr  Z– σB(x,r) gpdµ   1/p (2.3) 96 |Khanh Ngoc Nguyen and Zhuang Wang, Admissibility versus Ap-Conditions on Regular Trees where uB:=R −Bu dµ =1 µ(B)RBu dµ. We say that µis p-admissible if µis a doubling measure and (X,d,µ) supports a (1,p)-Poincaré inequality. The doubling property (2.2) and (1,p)-Poincaré inequality (2.3) can be defined on general metric measure spaces. In particular, on Rn, in view of [16, Theorem 2] or [15, Theorem 8.4.2], the (1,p)-Poincaré inequality (2.3) is equivalent to the (1,p)-Poincaré inequality given in the Introduction. It perhaps worth to point out that, since our K-regular trees are geodesic spaces, if µis p-admissible, the dilation constant σin (2.3) can be taken to 1, see [10] and [11]. 3Proof of Theorem 1.3 for K≥2 In this section, we give the proof of Theorem 1.3 for K≥2. To do so, we establish the following lemmas. Lemma 3.1. Let 1≤p<∞and Xbe a K-regular tree with distance dand measure µwhere K≥1. Assume that µsatisfies the Ap-condition. Then µis p-admissible. Proof. For 1≤p<∞, let CA:= sup Ap(x,r) : x∈X,r>0. Since µsatisfies the Ap-condition, 0<CA<∞. Case p= 1:We first show that µis a doubling measure. Let x∈Xand r>0be arbitrary. Notice that A1(x,2r)≤CA. Then it follows from (1.2) that ess supw∈[x,x2r] λ(w) Kj(w)−j(x)µ(w)≤4rCA µ(F(¯ x2r,4r)) . Hence r=Z [x,xr] ds =Z [x,xr] Kj(w)−j(x)µ(w) λ(w)!λ(w) Kj(w)−j(x)µ(w)ds(w) ≤  Z [x,xr] Kj(w)−j(x)µ(w) λ(w)ds(w)  4rCA µ(F(¯ x2r,4r)).(3.1) Notice that Z [x,xr] Kj(w)−j(x)µ(w) λ(w)ds(w) = µ(F(x,r)) ≤µ(B(x,r)) and that µ(F(¯ x2r,4r)) ≥µ(B(x,2r)). It follows from estimate (3.1) that r≤4CArµ(B(x,r)) µ(B(x,2r)) , which proves that µis a doubling measure with doubling constant 4CAsince r>0and the pair (x,r)is arbitrary. Next we prove that (X,d,µ)supports a (1,1)-Poincaré inequality. Consider an arbitrary ball B(x,r)with x∈Xand r>0. By the triangle inequality, we obtain that Z– B(x,r) |u−uB(x,r)|dµ ≤2Z– B(x,r) |u(y)−u(¯ xr)|dµ(y)(3.2) Khanh Ngoc Nguyen and Zhuang Wang, Admissibility versus Ap-Conditions on Regular Trees |97 for the left-hand side of our Poincaré inequality. By the definition of upper gradients and the Fubini theorem, for any upper gradient guof u, the right-hand side of (3.2) rewrites as 2Z– B(x,r) |u(y)−u(¯ xr)|dµ(y)≤2Z– B(x,r)Z [¯ xr,y] gu(w)ds(w)dµ(y) = 2 Z– B(x,r) gu(w)λ(w) µ(w)  Z B(x,r) χ[¯ xr,y](w)dµ(y)  dµ(w) = 2 Z– B(x,r) gu(w)λ(w) µ(w)µ({y∈B(x,r) : w∈[0,y]})dµ(w).(3.3) Here the last equality holds since χ[¯ xr,y](w)is not zero only if w∈[0,y]. Since the measure µsatisfies the A1-condition, A1(¯ xr,2r)<CA. It follows from (1.2) that µ(F(¯ x3r,4r)) 4r·ess supw∈[¯ xr,xr] λ(w) Kj(w)−j(¯ xr)µ(w)≤CA. Combining with the fact that Kj(¯ x3r)≤Kj(¯ xr), we obtain that λ(w) µ(w)µ({y∈B(x,r) : w∈[0,y]}) = λ(w) µ(w)Z {y∈[w,wr]∩B(x,r)} Kj(y)−j(w)µ(y) λ(y)ds(y) ≤λ(w)Kj(¯ x3r) µ(w)Kj(w)Z [¯ x3r,xr] Kj(y)−j(¯ x3r)µ(y) λ(y)ds(y) ≤λ(w) µ(w)Kj(w)−j(¯ xr)µ(F(¯ x3r,4r)) ≤4CAr(3.4) for any w∈B(x,r). Combining (3.2)-(3.4), yields Z– B(x,r) |u−uB(x,r)|dµ ≤8CArZ– B(x,r) gudµ for all balls B(x,r). Case p>1:Let us first prove that µis a doubling measure. Let B(x,r)be an arbitrary ball in X. Since µ satisfies the Ap-condition, we have Ap(x,2r)≤CA, and hence µ(F(¯ x2r,4r)) 4r·   1 2rZ [x,x2r] Kj(w)−j(x)µ(w) λ(w)!1 1−p ds(w)   p−1 ≤CA.(3.5) A simple calculation using the Hölder inequality shows that r=Z [x,xr] Kj(w)−j(x)µ(w) λ(w)!1/p Kj(w)−j(x)µ(w) λ(w)!−1/p ds(w) ≤  Z [x,xr] Kj(w)−j(x)µ(w) λ(w)ds(w)   1/p  Z [x,xr] Kj(w)−j(x)µ(w) λ(w)!1 1−p ds(w)   p−1 p ≤µ(F(x,r))1/p(2r)p−1 p   1 2rZ [x,x2r] Kj(w)−j(x)µ(w) λ(w)!1 1−p ds(w)   p−1 p . 98 |Khanh Ngoc Nguyen and Zhuang Wang, Admissibility versus Ap-Conditions on Regular Trees Inserting (3.5) into the above estimate yields r≤(2r)p−1 pµ(F(x,r))1/pµ(F(¯ x2r,4r)) 4rCA−1 p =CA1/p2p+1 prµ(F(x,r)) µ(F(¯ x2r,4r))1/p .(3.6) Note that µ(F(x,r)) ≤µ(B(x,r)) and µ(F(¯ x2r,4r)) ≥µ(B(x,2r)). Then the estimate (3.6) implies that r≤CA1/p2p+1 prµ(B(x,r)) µ(B(x,2r))1/p , which gives that µis a doubling measure with doubling constant CA2p+1, since r>0and B(x,r)is arbitrary. Next we show that (X,d,µ)supports a (1,p)-Poincaré inequality. Suppose B(x,r)is an arbitrary ball with center x∈Xand radius r>0. Since the measure µsatisfies the Ap-condition, then Ap(¯ xr,2r)<CA. It follows from (1.1) that µ(F(¯ x3r,4r)) 4r·   1 2rZ [¯ xr,xr] Kj(w)−j(¯ xr)µ(w) λ(w)!1 1−p ds(w)   p−1 ≤CA.(3.7) Recall that the left-hand side of our Poincaré inequality can be estimated by (3.3). A simple calculation shows that λ(w) µ(w)µ({y∈B(x,r) : w∈[0,y]}) = λ(w) µ(w)Z {y∈[w,wr]∩B(x,r)} Kj(y)−j(w)µ(y) λ(y)ds(y) ≤λ(w) µ(w)Kj(w)−j(¯ xr)Z [¯ xr,xr] Kj(y)−j(¯ xr)µ(y) λ(y)ds(y) =λ(w) µ(w)Kj(w)−j(¯ xr)µ(F(¯ xr,2r)) (3.8) for any point w∈B(x,r). Inserting the estimate (3.8) into (3.3) yields that Z– B(x,r) |u−uB(x,r)|dµ ≤2  Z– B(x,r) gu(w)λ(w) µ(w)Kj(w)−j(¯ xr)dµ(w)  µ(F(¯ xr,2r)). Applying the Hölder inequality for the right-hand side of the above inequality, it follows that Z– B(x,r) |u−uB(x,r)|dµ ≤2  Z– B(x,r) gupdµ   1/p  Z– B(x,r)λ(w) Kj(w)−j(¯ xr)µ(w)p p−1 dµ(w)   p−1 p µ(F(¯ xr,2r)).(3.9) By using the estimate (3.7), we obtain that   Z– B(x,r)λ(w) Kj(w)−j(¯ xr)µ(w)p p−1 dµ(w)   p−1 p µ(F(¯ xr,2r)) ≤µ(F(¯ xr,2r)) µ(B(x,r))p−1 p  Z F(¯ xr,2r)λ(w) Kj(w)−j(¯ xr)µ(w)p p−1 dµ(w)   p−1 p ≤µ(F(¯ xr,2r)) µ(B(x,r))p−1 p (2r)p−1 p   1 2rZ [¯ xr,xr] Kj(w)−j(¯ xr)µ(w) λ(w)!1 1−p ds(w)   p−1 p ≤µ(F(¯ xr,2r)) µ(B(x,r))p−1 p (2r)p−1 pµ(F(¯ x3r,4r)) 4rCA−1 p =CA1/p2p+1 prµ(F(¯ xr,2r)) µ(B(x,r))p−1 pµ(F(¯ x3r,4r))1/p .(3.10) Khanh Ngoc Nguyen and Zhuang Wang, Admissibility versus Ap-Conditions on Regular Trees |99 Note that F(¯ xr,2r)) ⊂B(x,4r)and that B(x,r)⊂F(¯ x3r,4r). Since µis a doubling measure with doubling constant CA2p+1, we have that µ(F(¯ xr,2r)) µ(B(x,r))p−1 pµ(F(¯ x3r,4r))1/p ≤µ(B(x,4r)) µ(B(x,r)) ≤(CA2p+1)2. Inserting the above estimate into the estimate (3.10), we have   Z– B(x,r)λ(w) Kj(w)−j(¯ xr)µ(w)p p−1 dµ(w)   p−1 p µ(F(¯ xr,2r)) ≤CA2+ 1 p2p+1 p+2(p+1)r.(3.11) Thanks to the estimates (3.9) and (3.11), we obtain Z– B(x,r) |u−uB(x,r)|dµ ≤CA2+ 1 p21 p+2p+4r  Z– B(x,r) gupdµ   1 p for all balls B(x,r). Lemma 3.2. Let 1≤p<∞and Xbe a K-regular tree with distance dand measure µwhere K≥2. Suppose that µis p-admissible. Then µsatisfies the Ap-condition. Proof. Let x∈Xand r>0be arbitrary. Let εbe an arbitrary positive number. Let x1∈Xbe a closest vertex of xwith |x1|>|x|. Then we define Tx1:= {y∈X:x1∈[0,y]}and T1:= [x,x1]∪Tx1 Since µis p-admissible, we may assume that µsatisfies the doubling condition (2.2) and the (1,p)-Poincaré inequality (2.3). Case p= 1:Let m=ess infw∈[x,xr 2] Kj(w)−j(x)µ(w) λ(w). In order to test the (1,1)-Poincaré inequality (2.3), we define u(y) =        0if y∈X\T1, R[x,y]χEε(w)ds(w)if y∈F(x,r/2) ∩T1, aotherwise where Eε:= nw∈F(x,r 2) : Kj(w)−j(x)µ(w) λ(w)<m+εoand a=R[x,xr 2]χEε(w)ds(w). Note that Eεis a non-empty set by the definition of mand that r>a=Z [x,xr 2] χEε(w)ds(w)>0. By the definition of u, we obtain that gu:= χEεis an upper gradient of u. Hence the right-hand side of the (1,1)-Poincaré inequality (2.3) is CPrZ– σB(x,r) gudµ =CPrZ– σB(x,r) χEε(w)dµ(w) =CPr µ(σB(x,r)) Z F(x,r/2) χEε(w)dµ(w) =CPr µ(σB(x,r)) Z [x,xr 2] χEε(w)Kj(w)−j(x)µ(w) λ(w)ds(w).