Khanh Nguyen
JYU DISSERTATIONS 417
Trace Operators and Classification
Criteria for Regular Trees
Khanh Nguyen
Trace Operators and Classification Criteria for Regular Trees
Esitetään Jyväskylän yliopiston matemaattis-luonnontieteellisen tiedekunnan suostumuksella julkisesti tarkastettavaksi elokuun 20. päivänä 2021 kello 12.
Academic dissertation to be publicly discussed, by permission of the Faculty of Mathematics and Science of the University of Jyväskylä,
on August 20, 2021 at 12 o’clock noon.
JYVÄSKYLÄ 2021
Editors Pekka Koskela
Department of Mathematics and Statistics, University of Jyväskylä Ville Korkiakangas
Open Science Centre, University of Jyväskylä
Copyright © 2021, by University of Jyväskylä
ISBN 978-951-39-8795-4 (PDF) URN:ISBN:978-951-39-8795-4 ISSN 2489-9003
Permanent link to this publication: http://urn.fi/URN:ISBN:978-951-39-8795-4 Jyväskylä University Printing House, Jyväskylä 2021
I wish to express my deepest gratitude for my supervisor Pekka Koskela for his skillful guidance, hospitality, endless patience, and support over the past years. I thank him for carefully reading and critically checking my drafts with many helpful suggestions and ideas which lead to significant improvements on my results. I am also extremely grateful to Jana Bj¨orn who found unnoticed mistakes in the paper [D] of this dissertation.
I thank the Department of Mathematics and Statistics at the Mattilanniemi campus for providing an excellent research environment and all the facilities one might wish for.
I want to say “Kiitos” to all the staff at our department for the warm and supporting atmosphere. I also want to thank my co-author Zhuang Wang for a pleasant and fruitful collaboration.
I am thankful to my bachelor’s and master’s thesis advisor Nguyen Thac Dung and my teacher Ninh Van Thu for encouraging me in my first steps of mathematical research.
Many thanks also to my colleagues from Department of Analysis at VNU Hanoi University of Science, Vietnam, for their help.
Finally, I want to thank my family and my friends for being part of my life.
Jyv¨askyl¨a, July 1, 2021 Khanh Nguyen
Abstract
The thesis deals with existence of traces of Sobolev functions defined on regular trees.
The existence is shown to be strongly related to the isoperimetric profile of the tree under natural assumptions. Furthermore, we give classification criteria for regular trees.
V¨ait¨oskirjassa tutkitaan Sobolev-funktioiden j¨alki¨a s¨a¨ann¨ollisten metristen puiden reu- noilla. J¨alkien olemassaolon osoitetaan liittyv¨an l¨aheisesti puun isoperimetriseen pro- fiiliin. Lis¨aksi annetaan kriteerej¨a s¨a¨ann¨ollisten metristen puiden luokitteluun.
List of included articles
This dissertation consists of an introductory part and the following four publications:
[A] Pekka Koskela, Khanh Nguyen, Zhuang Wang, Trace and density results on regular trees, to appear in Potential Analysis.
[B] Pekka Koskela, Khanh Nguyen, Zhuang Wang, Trace operators of regular trees, Analysis and Geometry in Metric Spaces 8 (2020), no. 1, 396–409.
[C] Khanh Nguyen, Zhuang Wang, Admissibility versus Ap-conditions on regular trees, Analysis and Geometry in Metric Spaces 8 (2020), no. 1, 92-105.
[D] Khanh Nguyen,Classification criteria for regular trees, accepted by Annales Academiæ Scientiarum Fennicæ. Mathematica.
The author of this dissertation has actively taken part in the research for the joint papers [A], [B] and [C].
In this thesis, we consider descriptions of traces of Sobolev functions on regular trees and classification criteria for regular trees. This will be done via an isoperimetric profile.
Let 1 < p < ∞ and let B ⊂Rn be the n-dimensional unit ball where n ≥2. Let us begin by considering a weight w ∈L1loc(Rn) and the associated measureµ,
(1.1) µ(E) =
Z
E
w(x)dx for each Borel subset E⊆B.
We say that u : B → R is a p-harmonic function if it satisfies the weighted p-Laplace equation
(1.2) −div(w(x)|∇u|p−2∇u) = 0 onB.
Solutions to this weighted p-Laplace equation can be recognized as local minimizers of the weighted p-Dirichlet integral
Z
B|∇u|pw(x)dx
and hence it follows that solutions are canonically attached to the weighted Sobolev space W1,p(B, µ).
If the weight w is p-admissible, that is doubling and supports a (1, p)-Poincar´e in- equality on B, then there exists a unique solution u ∈ W1,p(B, µ) of (1.2) such that u−v ∈ W01,p(B, µ) for a given v ∈W1,p(B, µ). Here W01,p(B, µ) is the closure of the col- lection of all functions u∈W1,p(B, µ) with compact support inB. Moreover, many basic regularity properties are also obtained under this admissibility assumption, see [3,19] for further information. It is well known that each MuckenhouptAp-weighted measure onRn isp-admissible, but the converse does not hold true forn≥2, see [7], [19, Section 15]. Sur- prisingly, on the real line R, anyp-admissible measure is actually given by anAp-weight, see [8]. 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. Since the local structure of a tree is one-dimensional, it is natural to study the relationship between p-admissibility and Muckenhoupt weights on trees. In Section 3, we will show that p-admissibility is equivalent to a version of a Muckenhoupt condition also on trees.
Motivated by boundary value problems for the weighted p-Laplace equation given by (1.2), we wish to know whether the limit
(1.3) lim
t→1u(tξ)
exists or not for Hn−1-a.e. ξ ∈S where S is the unit sphere. Here Hn−1 is the (n−1)- Hausdorff measure.
1
Let us discuss the unweighted case. Let u belong to the classical Sobolev space W1,1 on the unit open ball B, consisting of all integrable functions whose all first order distri- butional derivatives are also integrable over B. Then u has a representative v for which the limit (1.3) exists forHn−1-a.eξ in the unit sphere S.
Let us extend u to a function Eu ∈W1,1(Rn). This is possible by classical extension theorems in [10,40]. By the version of Lebesgue differentiation theorem for Sobolev functions [47, Section 5.14], the limit
r→0lim 1
|B(x, r)| Z
B(x,r)
Eu(y)dy
exists for Hn−1-a.e. x∈Rn. It then follows from the (1,1)-Poincar´e inequality that also
(1.4) lim
r→0
1
|B(x, r)∩B|
Z
B(x,r)∩B
u(y)dy
exists for Hn−1-a.e xand also that there is a valueT u(ξ) for which (1.5) lim
r→0
1
|B(ξ, r)∩B|
Z
B(ξ,r)∩B|u(x)−T u(ξ)|dx= 0 for Hn−1-a.e. ξ ∈S.
Thus we have three different possible traces, but it turns out that T u(ξ) coincides with the limits in (1.3) and (1.4) (for a suitable representativev) almost everywhere onS. Moreover, by the (q, p)-Poincar´e inequality (with 1≤q < ∞whenp≥nand 1≤q≤ npn−p when 1≤ p < n), we may replace the term |u(x)−T u(ξ)| by |u(x)−T u(ξ)|q in (1.5) if we assume that u∈W1,p(B).
Let us next consider a weightw when p >1.Suppose that (1.6)
Z
B|∇u(x)|pw(x)dx <∞ where w(x) =|x|1−n(1− |x|)p−1.
It is easy to check that the following radial function u satisfies (1.6) but limt→∞u(tξ) =
∞ for all ξ ∈S,
u(x) =k+ 1 2k
Z |x| ak
(1−t)−1dt
for |x| ∈ [ak, ak+1), k ∈ N, where 0 = a1 < a2 < . . . < 1 are chosen so that Rak+1
ak (1− t)−1dt= 2k.
We will study the existence of three different traces as above in the setting of trees.
In Section 4, we will introduce three different definitions of traces corresponding to three above traces and prove that they are equivalent. Moreover, we will give a characterization of the existence of an analog of the limit (1.3) on trees.
Finally, we consider classification problems. By the uniformization theorem of F. Klein, P. Koebe and H. Poincar´e for Riemann surfaces, every simply connected Riemann surface
2
half plane H2 (surface of hyperbolic type), the complex plane R2 (surface of parabolic type), the Riemann 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 toH2, otherwise we say that it is parabolic.
Let M be a simply connected Riemann surface with Riemannian metric g. A C2- smooth function u defined on M is superharmonic if −∆u≥ 0. Here ∆ is the Laplace- Beltrami operator associated to the Riemannian metric g. It is well known that every conformal mapping in dimension two preserves superharmonic functions (see [1, Page 135]). Since H2 possesses a nonconstant nonnegative superharmonic function and every nonnegative superharmonic function on R2 or on Sis constant, it then follows that there is no nonconstant nonnegative superharmonic function on (M, g) if and only if M is parabolic.
LetK be a compact subset in (M, g). We define the capacity Cap(K) by
Cap(K) = inf Z
M|∇u|2dmg :u∈Lip0(M), u|K ≡1
where Lip0(M) is a set of all Lipschitz functions with compact support on M, and mg is the Riemannian measure associated to g. Then there is a nonconstant nonnegative superharmonic function on M if and only if Cap(K) > 0 for some compact subset K, (see [14, Theorem 5.1] for Riemannian manifolds). It follows that the parabolicity of a Riemann surface M can be characterized both in terms of capacity and of superharmonic functions. By this reason, in the setting of Riemannian manifolds or metric measure spaces, one defines parabolicity either via capacity (see [23,25,27,28]) or via superharmonic functions (see [14] and also references therein).
We will study this problem in the setting of a tree. In Section 5, we will discuss the definition of parabolicity in terms of capacity and give characterizations of parabolicity.
The thesis is organized as follows.
In Section 2, we introduce some notation and concepts related to metric measure spaces, regular trees and their boundaries.
In Section 3, we give a characterization of p-admissibility on our trees.
In Section4, we introduce an isoperimetric profile. We define various trace operators on our trees and characterize the existence of the trace operators for first order Sobolev spaces on regular trees via our profile. Finally, we prove that the existence of all the various trace operators are equivalent.
In Section 5, we give characterizations for parabolicity of our trees.
3
2 Preliminaries
2.1 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 connected graph without cycles.
We call a treeG arooted tree if it has a distinguished vertex called theroot, which we will denote by 0. The neighbors of a vertex x∈ V are of two types: the neighbors that are closer to the root are called parents ofxand all other neighbors are calledchildren of x. Each vertex has a unique parent, except for the root itself that has none. We say that a tree G is K-regular if it is a rooted tree such that each vertex has exactly K children where K is a positive integer. Then all vertices except the root of G have degree K+ 1, and the root has degree K.
For simplicity of notation, we let X = V S
E and call it a K-regular tree. For any curve γ, let lG(γ) be the metric graph length of γ given by
lG(γ) = Z
γ
dG
wheredGis the length element onG. A geodesic connectingx, y∈X is denoted by [x, y].
We notice that [x, y] is unique for given x, y ∈ X. We denote by |x| the metric graph length of the unique geodesic [0, x]. Then the metric graph distance between two vertices is the number of edges needed to connect them. We say that γ is an infinite geodesic in X ifγ is a simple curve and lG(γ) =∞.
A tree is the quintessential Gromov hyperbolic space, and hence we can consider the visual boundary of the tree as in Bridson-Haefliger [9]. We define the boundary of our K-regular tree X, denoted ∂X, as the collection of all infinite geodesics in X starting at the root 0. Then every ξ ∈ ∂X corresponds to an infinite geodesic [0, ξ) that is an isometric copy of the interval [0,∞). Given two points ξ, ζ ∈ ∂X, there is an infinite geodesic (ξ, ζ) inX connectingξ and ζ.
To avoid confusion, points inX are denoted by Latin letters such asx,y, andz, while for points in ∂X we use Greek letters such as ξ, ζ, and η.
2.2 Metric and measure
Let X be a K-regular tree as in Section 2.1 for some positive integer K. We define a metric d via dλ and a measure µw by setting
(2.1) dλ(x) =λ(|x|)dG(x) and dµw(x) =w(|x|)dG(x) 4
points x, y∈X, the distance between xand y, denoted d(x, y), is d(x, y) =
Z
[x,y]
dλ = Z
[x,y]
λ(|z|)dG(z)
where [x, y] is the unique geodesic betweenx, y. Ifx∈[0, y] then the distance betweenx and y is given by
d(x, y) = Z |y|
|x|
λ(t)dt.
We say that d is an infinite metric if the diameter ofX is infinite, otherwise d is said to be finite. For any Borel subset Aof X, the measure of A, denoted µw(A), is
µw(A) = Z
A
dµw= Z
A
w(|x|)dG(x).
The measure of ourK-regular tree is µw(X) =
X∞ i=0
Z ∞
0
w(|x|)χ{x:i≤|x|≤i+1}(x)dG(x) = Z ∞
0
Kj(t)w(t)dt
where j(t) is the largest integer such that j(t) ≤ t+ 1. We say that µw is an infinite measure if µw(X) =∞, otherwise it is said to be finite.
We abuse the notation and let w(x) and λ(x) denote w(|x|) and λ(|x|), respectively, for every x∈X if there is no danger of confusion.
Theorem 2.1 (Theorem 2.1 in [D]). The identity mapping from (X, dG) to (X, d) is a homeomorphism.
Let Xn = {x ∈ X : |x| ≤ n} for n ∈ N. As Xn is compact in (X, dG) for each n ∈N and any compact set in (X, dG) is contained in some Xn, Theorem 2.1gives that (X, d, µw) is a connected, locally compact, and non-compact metric measure space.
Let us close this section to introduce a metric db and a measure ν on the boundary
∂X for a given K-regular tree X with metric d and measureµw as above. We denote by (ξ, ζ) the infinite geodesic connectingξ and ζ. Then (ξ, ζ) consists of the tails [x, ξ) and [x, ζ) of the geodesics [0, ξ) and [0, ζ) starting at the last common point x of [0, ξ) and [0, ζ). We define the visual metric db on∂X, see [9] for more details, by setting
db(ξ, ζ) = Z
(ξ,η)
dλ = 2 Z ∞
|x(ξ,ζ)|
λ(t)dt
for any ξ, ζ ∈ ∂X where x(ξ,ζ) is the last common point of [0, ξ) and [0, ζ). Then the metric space (∂X, db) is an ultrametric space for all K-regular trees X with finite metric d, see [B, Proposition 2.1].
5
We equip ∂X with the natural measure ν as in Falconer [11] by distributing the unit mass uniformly on ∂X. Then the boundary measure of A, denotedν(A), is
ν(A) = Z
A
dν
for any Borel subset A ⊂ ∂X. For any x ∈ X with |x| = j, we have ν(Γx) = K−j for Γx := {ξ ∈ ∂X : x ∈ [0, ξ)}. We refer to [4, Lemma 5.2] for more information on our boundary measure ν.
2.3 Modulus of a curve family
Let 1 ≤ p < ∞ and let X be a K-regular tree with metric d and measure µw as in (2.1). We denote the family of all nonconstant (locally) rectifiable curves in X by M. Given a subfamily Γ ⊂ M, we define the p-modulus of the family Γ, denoted Modp(Γ), by setting
Modp(Γ) = inf Z
X
ρpdµw
where the infimum is taken over all Borel-measurable functions ρ:X →[0,∞] such that Z
γ
ρdλ ≥1 for every γ ∈Γ.
Theorem 2.2 (Proposition 2.2 in [A]). Let1≤p <∞. Then the following are equivalent:
1. λ/w1/p ∈Lploc∗([0,∞)) where p∗ is the H¨older conjugate of p, that is 1p +p1∗ = 1.
2. Modp({γ})> 0for any rectifiable curve γ.
In particular, on our K-regular tree X with metric and measure as in Section 2.1, the empty family is the only curve family with zero p-modulus for 1≤p <∞.
2.4 Newtonian spaces
Let f be a locally integrable function on (X, d, µw). We say that a Borel function g :X →[0,∞] is an upper gradient off if
(2.2) |f(x)−f(y)| ≤
Z
γ
g dλ
whenever x, y ∈X and γ is the unique geodesic connecting x and y. For our K-regular trees, any (locally) rectifiable curve with end pointsxand y contains the unique geodesic [x, y] and hence the upper gradient defined above is equivalent to the definition which requires that the inequality (2.2) holds for all (locally) rectifiable curves with end points
6
g :X →[0,∞] is called ap-weak upper gradient ofuif (2.2) holds onp-a.e curveγ ∈ M, i.e there exists a subfamily Γ⊂ M with Modp(Γ) = 0 such that (2.2) holds for all curves γ ∈ M \Γ. By Theorem 2.2, any p-weak upper gradient is actually an upper gradient in our setting. The notion of upper gradients is due to Heinonen and Koskela [21]. We refer interested readers to [3,16,22,41] for a more detailed discussion on p-weak upper gradients.
Given 1 ≤ p < ∞, the Newtonian space N1,p(X) := N1,p(X, d, µw) is defined as the collection of all functions uwith finite N1,p-norm
kukN1,p(X):=kukLp(X)+ inf
g kgkLp(X)
where the infimum is taken over all upper gradients g of u. If u ∈ N1,p(X) then it has a minimal p-weak upper gradient, which is an upper gradient in our case by Theorem 2.2. We denote by gu the minimal upper gradient, which is unique up to measure zero and which is minimal in the sense that if g ∈ Lp(X) is any upper gradient of u then gu ≤ g a.e. Throughout this thesis, we denote by gu the minimal upper gradient of u.
We refer to [16, Theorem 7.16] for proofs of the existence and uniqueness of such minimal upper gradient. By Theorem 2.2 and [16, Definition 7.2 and Lemma 7.6], it follows that any function u ∈ L1loc(X) with an upper gradient 0 ≤ g ∈ Lp(X) is locally absolutely continuous, for example, absolutely continuous on each edge. Moreover, the classical derivative u0 of this locally absolutely continuous function is a minimal upper gradient in the sense that gu(x) =|u0(x)|/λ(x) when uis parameterized in the natural way.
We define the homogeneous Newtonian spaces N˙1,p(X), 1 ≤p <∞, the collection of all continuous functions u that have an upper gradient 0 ≤ g ∈ Lp(X), for which the homogeneous ˙N1,p-norm ofu defined as
kukN˙1,p(X):=|u(0)|+ inf
g kgkLp(X)
is finite. Here 0 is the root of the K-regular tree and the infimum is taken over all upper gradients g of u.
2.5 Doubling and Poincar´ e inequalities
Let λ, w: [0,∞)→ (0,∞) be locally integrable Borel-functions. We associate a metric d and a measure µw to λ and w as in Section 2.2. We say that µw is doubling on (X, d) if there exists a positive constant CD such that for all ballsB(x, r) :=Bd(x, r) with radius r > 0 and center atx∈X,
µw(B(x,2r))≤CDµw(B(x, r)) where the constant CD is called the doubling constant.
7
Let 1 ≤ p < ∞. Our space (X, d, µw) supports a (1, p)-Poincar´e inequality if there exist positive constantsCP >0 andσ≥1 such that for all ballsB(x, r) with radiusr >0 and center at x∈X, every integrable function uonσB(x, r) and all upper gradientsg of u,
− Z
B(x,r)|u−uB(x,r)|dµw ≤CPr
− Z
σB(x,r)
gpdµw 1p
where uB :=R−
Budµw = µ(B)1 R
Budµw and σB(x, r) :=B(x, σr).
3 Admissible weights
Let X be a regular tree with metric d and measure µw as in Section 2.2. We say that w is p-admissible if its associated measure µw is doubling and supports a (1, p)-Poincar´e inequality. In Rn, the class of p-admissible weights for Sobolev spaces and differential equations was introduced in [19]. The definition was initially based on four conditions, after which Theorem 2 in [17], Theorem 5.2 in [20], and Section 20 in [19] reduced them to the two conditions of doubling and (1, p)-Poincar´e inequality.
In order to give a characterization forp-admissibility, we need the following concepts.
For anyx∈Xandr >0, we denote by ¯xrthe point in [0, x] withd(¯xr, x) = min{r, d(0, x)} and denote by xr a point in X such that x ∈ [0, xr] with d(xr, x) = r. Hence ¯xr is the unique ancestor of xand xr is a descendant of x. Let
F(x, r) ={y ∈X :x∈[0, y], d(x, y)< r} be the downward directed “half ball”. We set
Ap(x, r) = µw(F(¯xr,2r))
2r · 1
r Z
[x,xr]
Kj(y)−j(x)w(y) λ(y)
1−p1 dλ(y)
!p−1
if 1< p <∞, and
A1(x, r) = µw(F(¯xr,2r))
2r ·ess sup
y∈[x,xr]
λ(y) Kj(y)−j(x)w(y)
where j(x) and j(y) are the largest integers such that j(x) ≤ |x|+ 1 and j(y) ≤ |y|+ 1 respectively. Notice that Ap(x, r) is independent of the choice ofxr among the points z with x∈[0, z] andd(x, z) =r.
Let 1≤p <∞ and X beK-regular with metric dand measure µw. We say that w is a MuckenhouptAp-weight if
sup{Ap(x, r) :x∈X, r >0}< ∞. 8
sup{Ap(x, r) :x∈X,0< r≤8d(0, x)}< ∞.
The following theorem is a characterization ofp-admissible weights on our tree.
Theorem 3.1 (Theorem 1.3 in [C]). Let 1 ≤ p < ∞ and X be a K-regular tree with infinite metric d and measure µw as in (2.1). Then
1. ForK = 1, w is p-admissible if and only ifw is a Muckenhoupt Ap-weight far from 0.
2. For K ≥2, w is p-admissible if and only if w is a MuckenhouptAp-weight.
4 Trace operators
In this section, we give a characterization of the existence of radial traces for first order Sobolev spaces defined on regular trees. Moreover, we show that the existence of radial traces is equivalent to the existence of two other traces. Throughout this section, we always assume that X is aK-regular tree with metricdand measureµw as in (2.1) where K ≥1.
We first introduce three definitions of traces on our tree. Given ξ ∈ ∂X, we refer to points x∈[0, ξ) by xξ.
Definition 4.1. Let f be a function defined on X. If the limit limxξ→ξf(xξ) exists at ξ ∈∂X, we write
(4.1) TRf(ξ) = lim
xξ→ξf(xξ)
and call it the radial trace of f at ξ. If TRf(ξ) exists for ν-a.e ξ ∈∂X, then we say that the radial trace TRf exists.
Towards defining analogs of (1.4) and (1.5), we set
Γx:={y∈X :x∈[0, y]} for givenx∈X.
Notice that Γx is also aK-regular tree ifx is a vertex, obviously with rootx.
Definition 4.2. Let 1≤q <∞. We assume that µw(X)<∞. Fix a function f defined on X. We say that the Lebesgue-point-type trace TL of f on ∂X exists if
TLf(ξ) := lim
xξ→ξ
1 µw(Γxξ)
Z
Γxξ
f(y)dµw(y) exists for ν-a.e ξ ∈∂X.
We say that the boundary trace of f of order q on ∂X exists if there is a function Tqf :
∂X →R so that
xlimξ→ξ
1 µw(Γxξ)
Z
Γxξ|f(y)−Tqf(ξ)|qdµw(y) = 0 for ν-a.e ξ ∈∂X.
9
One can find versions of the notions of traces in Definition4.2in literature under various names. We refer the readers to [15, Chapter 2], [36, Section 6.6], [37, Section 9.6], [47, Section 3.1] for discussions in the setting of Euclidean spaces, and [33–35] for discussions in the setting of metric measure spaces. Notice that in the setting of a MuckenhouptAp- weight discussed in our introduction, the analogs of the traces TRf, TLf, Tqf,1≤q ≤ p, exist and actually coincide with each other almost everywhere on S.
Classical trace results on the Euclidean spaces can be found in [2,12,18,30,32,38,39,43, 44]. For the case of manifolds see [23,24] and for the setting of metric spaces see [3,6,31].
In order to give a characterization for the existence of the limit (4.1) for functions in homogeneous Newtonian spaces, we need more concepts. Given 1 < p <∞, we define
Rp = Z ∞
0
λp−1p (t)w1−p1 (t)K1−pj(t)dt and
R1 = λ(t)
w(t)Kj(t)
L∞([0,∞))
where j(t) is the largest integer such thatj(t)≤t+ 1.
For 1 < p < ∞, actually Rp = ∞ guarantees parabolicity (every compact set is of relative p-capacity zero), see Section5for further discussion.
Our first result of this section gives a rather complete solution for the existence of radial traces TRf (4.1) for ˙N1,p-functions.
Theorem 4.3 (Theorem 1.3 in [A]). Let 1≤p <∞. Then the following are equivalent:
1. Rp <∞.
2. The radial trace TRf exists for every f ∈N˙1,p(X).
3. The radial trace TRf exists for every f ∈ N˙1,p(X) and TR : ˙N1,p(X)→ Lp(∂X, ν) is a bounded linear operator.
4. ˙N01,p(X)(N˙1,p(X). Here N˙01,p(X) is the collection of all N˙1,p-functions with com- pact support.
Our second result is an analog of the preceding result forN1,p(X)-functions in the case of finite measure.
Theorem 4.4 (Theorem 1.1 in [A]). Let 1 ≤p < ∞. Suppose that µw(X)< ∞. Then the following are equivalent:
1. Rp <∞.
2. The radial trace TRf exists for every f ∈N1,p(X).
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is a bounded linear operator.
4. N01,p(X)(N1,p(X). Here N01,p(X) is the collection of all N1,p-functions with com- pact support.
To deal with the limit (4.1) forN1,p-functions in the case of infinite volume, we intro- duce one more quantity, denoted Rp, by setting for 1< p <∞,
Rp = sup
k≥1
Z
Ak
λp−1p (t)w1−p1 (t)K1−pj(t)dt and R1 =R1
where {Ak}∞k=1:={[tk, tk+1)}∞k=1 is the sequence of subintervals in [0,∞) with Z
Ak
Kj(t)w(t)dt≡1 for k ∈N and 0< t1 < t2 < . . .. Then [0,∞) =S∞
k=1Ak.
Our third result completes the answer for the case ofN1,p-functions.
Theorem 4.5 (Theorem 1.2 in [A]). Let 1 ≤p < ∞. Suppose that µw(X) =∞. Then the following hold:
a) N01,p(X) =N1,p(X).
b) The following are equivalent 1. Rp <∞.
2. The radial traceTRf exists for every f ∈N1,p(X).
3. The radial traceTRf exists and TRf ≡0 for every f ∈N1,p(X).
Finally, we have that the existence of any of TRf, TLf, Tqf,1 ≤ q ≤ p, for all f ∈ N1,p(X) is equivalent to the finiteness of Rp. Moreover, all these different traces of f coincide when Rp <∞.
Theorem 4.6 (Theorem 1.3 in [B]). Let 1 ≤ q ≤ p < ∞. Suppose that µw(X) < ∞. Then the following are equivalent:
1. TRf exists for every f ∈N1,p(X).
2. TLf exists for every f ∈N1,p(X).
3. Tqf exists for every f ∈N1,p(X).
4. Rp <∞.
Moreover, if one of TRf, TLf, Tqf exists for each f ∈ N1,p(X), then all of them exist and concide ν-a.e on ∂X for a given f.
11
5 Classification problems
In this section, if we do not specifically mention, we always assume that 1 < p <∞ and that X is aK-regular tree with metricd and measureµw as in (2.1).
LetO be a subset of X. We define thep-capacity ofO, denoted Capp(O), by setting (5.1) Capp(O) = inf
Z
X
gupdµw :u|O ≡1, u∈N01,p(X)
wheregu is the minimal upper gradient ofuas in Section2.2. AK-regular tree X is said to bep-parabolic if Capp(O) = 0 for all compact setsO ⊂X; otherwiseX isp-hyperbolic.
Given an open subset Ω⊆X, we say thatu∈Nloc1,p(Ω) is a p-harmonic function (or a p-superharmonic function) on Ω if
(5.2)
Z
spt(ϕ)
gupdµw ≤ Z
spt(ϕ)
gu+ϕp dµw
holds for all functions (or for all nonnegative functions)ϕ∈N1,p(Ω) with compact support spt(ϕ)⊂Ω. We refer the interested readers to [3,19,22] for a discussion on thep-capacity and p-(super)harmonic functions.
It is natural to ask whether the parabolicity (or hyperbolicity) of X can be charac- terized via p-(super)harmonic functions under some conditions on the measure µw only depending on the given metric d, and also to ask for intrinsic conditions of K-regular trees that would characterize the parabolicity (or hyperbolicity). We refer the readers to [1, Chapter IV] for a discussion in the case of Riemann surfaces, and [13,14,25,27,28]
for a discussion in the setting of Riemannian manifolds, and [45, Section 6], [46] for a discussion on infinite networks.
The first result of this section is a characterization of parabolicity of K-regular trees.
Theorem 5.1 (Theorem 1.1 in [D]). Let K ≥ 1. Then (X, d, µw) is p-parabolic if and only if any one of the following conditions is fulfilled:
1. Rp =∞.
2. Capp(Xn) = 0 for all n∈N∪ {0}. 3. Capp(Xn) = 0 for some n∈N∪ {0}.
Recall that the condition Rp < ∞ gives a characterization of the existence of trace operators and for density properties of ˙N1,p(X) in Theorem 4.3. Hence parabolicity of K-regular trees can be characterized in terms of trace operators and density properties.
It is well known, see for instance the survey paper [27], that the volume growth condi-
tion Z ∞
1
t V(B(0, t))
p−11
dt=∞ 12
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 [25] and to Varopoulos [42] in the case p= 2. Our condition Rp = ∞ is an analog of this volume growth condition. The following example shows that there exists a K-regular tree with a distance dand a “non-radial” measureµw such thatRp =∞ butX is p-hyperbolic.
Example 5.2 (Example 3.8 in [D]). For simplicity, let X be a dyadic tree (which means K = 2). Then the root 0 of our tree has two closest vertices, denoted v1 and v2. We denote
T1 = [0, v1]∪ {x∈X :v1 ∈[0, x]} and T2 = [0, v2]∪ {x∈X :v2 ∈[0, x]}.
Notice that the union of T1 and T2 is our tree. Let λi, wi : [0,∞)→ (0,∞), for i= 1,2.
We introduce a measure µw and a metric d via the graph-length element dG by setting dµw(x) =wi(|x|)dG(x), dλ(x) =λi(|x|)dG(x),
for all x ∈ Ti, for i = 1,2. We choose λ1 ≡ w1 ≡ 1 and λ2 ≡ 1, w2(x) = 2−j(x). Then Rp =∞ but X is p-hyperbolic.
Let 1< p <∞. Set int(Xn) = {x ∈X : |x| < n} when n ∈N. Then (int(Xn), d, µ) is said to be doubling and to support a p-Poincar´e inequality if there exist constants C1≥1, C2 >0 only depending onn such that for all balls B(x,2r)⊂int(Xn),
µ(B(x,2r))≤C1µ(B(x, r)) and for all balls B(x, r)⊂int(Xn),
− Z
B(x,r)
|u−uB(x,r)|dµ≤C2r
− Z
B(x,r)
gpdµ p1
whenever u is a measurable function on B(x, r) andg is an upper gradient ofu.
Our second result of this section deals with a characterization of parabolicity in terms of p-(super)harmonic functions.
Theorem 5.3(Theorem 1.3 in [D]). LetK≥2. Assume additionally that(int(Xn), d, µw) is doubling and supports a p-Poincar´e inequality for each n ∈ N. Then (X, d, µw) is p- parabolic if and only if any one of the following conditions is fulfilled:
1. Every nonnegative p-superharmonic function u on X is constant.
2. Every nonnegative p-harmonic function u on X is constant.
13
3. Every bounded p-harmonic function u on X is constant.
4. Every bounded p-harmonic function u on X with R
Xgpudµw <∞ is constant.
According to a version of Theorem 5.3 in the setting of Riemannian manifolds from [23,26,29] we have that 1. ⇒ 2. ⇒ 3. ⇒ 4. However 3. does not imply 2. in gen- eral.
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17
Included articles
Trace and density results on regular trees P. Koskela, K. Nguyen, and Z. Wang
To appear in Potential Anal.
Potential Analysis
https://doi.org/10.1007/s11118-021-09907-2
Trace and Density Results on Regular Trees
Pekka Koskela1·Khanh Ngoc Nguyen1 ·Zhuang Wang2
Received: 25 February 2020 / Accepted: 28 January 2021 /
©The Author(s) 2021
Abstract
The boundary of a regular tree can be viewed as a Cantor-type set. We equip our tree with a weighted distance and a weighted measure via the Euclidean arc-length and consider the associated first-order Sobolev spaces. We give characterizations for the existence of traces and for the density of compactly supported functions.
Keywords Regular tree·Boundary trace·Newtonian space·Density Mathematics Subject Classification (2010) 46E35·30L99
1 Introduction
LetGbe aK-regular tree with a set of verticesV and a set of edgesEfor someK ≥1. The union ofV andE will be denoted byX. We abuse the notation and callXaK-regular tree.
We introduce a metric structure onXby considering each edge ofXto 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. Ifx is a vertex, we define|x|to be the distance between 0 andx. Since each edge is an isometric copy of the unit interval, we may extend this distance naturally to any x belonging to an edge. We define∂X as the collection of all infinite geodesics starting at
All authors have been supported by the Academy of Finland via Centre of Excellence in Analysis and Dynamics Research (project No. 307333). This work was also partially supported by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund.
Pekka Koskela pekka.j.koskela@jyu.fi Khanh Ngoc Nguyen khanh.n.nguyen@jyu.fi Zhuang Wang
zhuang.z.wang@foxmail.com; wangzhuangzhao@gmail.com
1 Department of Mathematics and Statistics, University of Jyv¨askyl¨a, PO Box 35, FI-40014 Jyv¨askyl¨a, Finland
2 LCSM(MOE), School of Mathematics and Statistics, Hunan Normal University, 410081, Changsha, China
the root 0. Then everyξ ∈ ∂X corresponds to an infinite geodesic[0, ξ ) (in X) that is an isometric copy of the interval[0,∞). Hencex →ξ along[0, ξ )has a canonical meaning.
Given a functionf defined onX,we are interested in the collection of thoseξ ∈∂Xfor which the limit off (x) exists whenx → ξ along [0, ξ ). We begin by equipping∂X with the natural probability measureνso thatν(Ix)=K−j whenx is a vertex with|x| =j and Ix = {ξ ∈∂X : x ∈ [0, ξ )}. Towards defining the classes of functions that are of interest to us, we define a measure function and a new distance function onX. Writed|x|for the length element onX and letμ : [0,∞) → (0,∞)be a Borel measurable and locally integrable function. We abuse notation and refer also to the measure generated viadμ(x)=μ(|x|)d|x| by μ. Further, let λ : [0,∞) → (0,∞) be Borel measurable and locally integrable, and we define a distanced viads(x) = λ(|x|)d|x| by settingd(z, y) =
[z,y]ds(x)whenever z, y ∈ X and [z, y]is the unique geodesic between z andy. For convenience, we assume additionally thatλp/μ∈ L1/(ploc −1)([0,∞)) ifp > 1 below and thatλ/μ ∈ L∞loc([0,∞))if p = 1. Then(X, d, μ) is a metric measure space and we letN1,p(X) := N1,p(X, d, μ), 1 ≤p <∞,be the associated Sobolev space based on upper gradients [10], as introduced in [27]. See Section 2 for the precise definition. We show there that, actually, each u ∈ N1,p(X)is absolutely continuous on each edge, withu ∈Lpμ(X). As usual,N01,p(X)is the completion of the family of functions with compact support inN1,p(X).
In order to state our results, we need two more concepts. Given 1< p <∞we set Rp =
∞
0
λ(t)p−1p μ(t)1−p1 K1−pt dt =
λ(x) μ(x)K|x|
p p−1
L
p−1p
μ (X)
and we define
R1 = λ(t)
μ(t)Kt
L∞([0,∞))
.
One should viewRp as an isoperimetric profile(X, d, μ) :in case of a Riemannian mani- foldM, the natural version ofRp is closely related to the parabolicity of the manifold [28];
Rp = ∞ guarantees parabolicity (every compact set is of relative p-capacity zero). This suggests that the existence of limits for Sobolev functions along geodesics might be some- how related to finiteness ofRp, see Remark 3.8 for a discussion. Let us say that the trace of a given functionf, defined onX, exists if
Trf (ξ ):= lim
[0,ξ )x→ξf (x) (1.1)
exists forν-a.e.ξ ∈∂X. We then denote by Trf the trace function off. For other possible definitions of the trace and connections between them see [17].
Our first result gives a rather complete solution for the existence of traces in the case μ(X) <∞.
Theorem 1.1 LetXbe aK-regular tree with distanced and measureμ. Assumeμ(X) <
∞. For1≤p <∞, the following are equivalent:
(i) Rp <∞.
(ii) Trf exists for everyf ∈ N1,p(X)and Tr :N1,p(X)→Lpν(∂X)is a bounded linear operator.
(iii) Trf exists for everyf ∈ N1,p(X).
(iv) N01,p(X)N1,p(X).
Trace and Density Results on Regular Trees
In [3,18] the trace spaces of our Sobolev spaces were identified as suitable Besov-type spaces for very specific choices ofμ, λ.
For the caseμ(X)= ∞, we define Rp =sup
k≥1
Ak
λ(t)
p−1p μ(t)1−p1 K1−pt dt ifp >1 and R1=R1 (1.2) where {Ak}∞k=1 = {[tk, tk+1)}∞k=1 is the sequence of subintervals in [0,∞) with
AkKtμ(t)dt =1 fork =1,2, . . .and 0=t1 < t2 < . . . Then[0,∞)=∞
k=1Ak. Our second result deals with the case of infinite volume.
Theorem 1.2 Let1 ≤p < ∞. Let Xbe aK-regular tree with distanced and measureμ.
Assumeμ(X)= ∞. Then the following hold:
(1) N01,p(X)=N1,p(X).
(2) The following are equivalent:
(a) Rp <∞.
(b) Trf exists and Trf ≡0for everyf ∈N1,p(X).
(c) Trf exists for everyf ∈N1,p(X).
In this case of infinite volume and 1 < p < ∞, it is easy to see thatRp < ∞implies Rp < ∞,but the inverse implication need not hold true. Moreover, the finiteness of Rp does not imply the finiteness ofRq for some 1 ≤ q < ∞, see Example 3.14, and Remark 3.15.
Our third result gives a complete answer in the case of homogeneous norms, see Section2 for the relevant definitions. HereN˙01,p(X)is the completion of the family of functions with compact support inN˙1,p(X).
Theorem 1.3 LetXbe aK-regular tree with distanced and measureμ. For1≤p < ∞, the following are equivalent:
(i) Rp <∞.
(ii) Trf exists for everyf ∈ ˙N1,p(X)and Tr : ˙N1,p(X)→Lpν(∂X)is a bounded linear operator.
(iii) Trf exists for everyf ∈ ˙N1,p(X).
(iv) N˙01,p(X)N˙1,p(X).
Let us close this introduction with some comments on Theorem 1.3. Even though the condition Rp = ∞impliesp-parabolicity, finiteness of this quantity does not, in general, prevent p-parabolicity, see [7]. Hence Theorem 1.3 and the preceding theorems are some- what surprising. In fact, it follows from our results that, in the setting of this paper,Rp = ∞ precisely when(X, d, μ)isp-parabolic. See [23] for more on this. Hence the reader famil- iar with moduli of curve families might wish to view Theorem 1.3 as kind of a version of the equivalence between modulus and capacity.
Partial motivation for this paper comes from boundary value problems for thep-Laplace equation. For the case of manifolds see [12,13] and for the setting of metric spaces see [2, 4,15]. Classical trace results on the Euclidean spaces can be found in [1,6,9,14,16,22, 25,29,30] and studies of parabolicity on infinite networks in [26,32]. For trace results in the metric setting see [3,18–21,33]. Our second motivation comes from the recent paper [24] where a version of Theorem 1.3 was established on regular trees for the casep =2.
The paper is organized as follows. In Section2, we introduce regular trees, boundaries of trees and Newtonian spaces on our trees. We study the trace results in Section3and the density results are given in Section4. In Section5, we give the proofs of Theorems 1.1–1.3.
2 Preliminaries
Throughout this paper, the letterC (sometimes with a subscript) will denote positive con- stants that usually depend only on the space and may change at different occurrences; ifC depends ona, b, . . ., we writeC = C(a, b, . . .). The notationA B (A B) means that there is a constantC such thatA≤C·B(A ≥C·B). The notationA≈B means that both A B andB Ahold. For any functionf ∈ L1loc(X)and any measurable subsetA ⊂X withμ(A) >0, we denote μ(A)1
Af dμby−
Af dμ.
2.1 Regular Trees and Their Boundaries
A graphGis 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. AtreeGis a connected graph without cycles.
We call a treeGarooted treeif it has a distinguished vertex called the root, which we will denote by 0. 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.
We say that a treeGisK-regular ifGis a rooted tree such that each vertex has exactly K children for someK ≥1. Then all vertices except the root ofGhave degreeK +1, and the root has degreeK.
LetGbe aK-regular tree with a set of verticesV and a set of edgesEfor someK ≥1.
For simplicity of notation, we letX= V ∪Eand call it aK-regular tree. AK-regular tree X is made into a metric graph by considering each edge as a geodesic of length one. For x ∈ X, let|x|be the distance from the root 0 tox, that is, the length of the geodesic from 0 tox, where the length of every edge is 1 and we consider each edge to be an isometric copy of the unit interval. The geodesic connectingx, y ∈ V is denoted by[x, y], and its length is denoted by|x−y|. If |x| < |y| andx lies on the geodesic connecting 0 toy, we write x < y and call the vertexya descendant of the vertexx. More generally, we writex ≤yif the geodesic from 0 toypasses throughx, and in this case|x−y| = |y| − |x|.
On theK-regular treeX, for anyn∈N, letXn be a subset ofXby setting Xn := {x ∈ X: |x|< n}.
On theK-regular treeX, we define a metricdviads and measuredμby setting dμ=μ(|x|) d|x|, ds(x)=λ(|x|) d|x|,
whereλ, μ : [0,∞) →(0,∞)are Borel functions withλ, μ ∈ L1loc([0,∞)). Throughout this paper, we let 1 ≤ p < ∞ and assume additionally that λp/μ ∈ L1/(ploc −1)([0,∞)) if p > 1 and that λ/μ ∈ L∞loc([0,∞)) if p = 1. Hered|x| is the measure which gives each edge Lebesgue measure 1, as we consider each edge to be an isometric copy of the
