Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 44, 2019, 281–291

ABSOLUTELY CONTINUOUS FUNCTIONSON COMPACT AND CONNECTED
1-DIMENSIONAL METRIC SPACES
Xiaodan Zhou
Worcester Polytechnic Institute, Mathematical Sciences Department
Worcester, MA 01609-2280, U.S.A.; xzhou3@wpi.edu
Abstract. In this paper, we study the absolutely continuous characterization of Sobolev
functions on compact and connected 1-dimensional metric spaces X. We generalize the definitionof absolutely continuous functions to such spaces and prove the equivalence between the absolutely
continuous functions and Newtonian Sobolev functions. We also show that a compact and 1Ahlfors regular metric space X supports a p-Poincaré inequality for 1 ≤ p ≤ ∞ if and only if
X is quasiconvex. As a result, the absolutely continuous functions are equivalent to the Sobolev
functions defined via several different approaches.

1. Introduction
On the real line, the absolutely continuous function u on a compact interval [a, b]
is defined in the following way: for all ǫ > 0, there exists a δ > 0 such that for any
finite collection of pairwise disjoint subintervals [ai , bi ] ⊂ [a, b],X|u(bi ) − u(ai )| < ǫPiprovided that i |bi − ai | < δ. We denote the absolutely continuous functions on[a, b] by AC([a, b]). An absolutely continuous function f ∈ AC([a, b]) is differentiable
almost everywhere. Furthermore, if the derivative f ′ is p-integrable, this function fbelongs to the Sobolev space W 1,p ([a, b]). On the other hand, every Sobolev function in W 1,p ([a, b]) has an absolutely continuous representative. Upon choosing sucha representative, we can identify a Sobolev function u ∈ W 1,p ([a, b]) with an absolutely continuous function in AC([a, b]) with p-integrable derivative [5]. In thiswork, we extend the definition of absolutely continuous functions to some general
one-dimensional metric spaces and study the connection between these absolutely
continuous functions and Sobolev functions.
There are many extensions of the definitions of absolutely continuous functions togeneral settings and some results on the connections between the absolutely continuous functions and Sobolev functions. Malý [14] introduces a class of “n-absolutely
continuous functions” in Rn and proves that absolute continuity implies continuity, weak differentiability with gradient in Ln and some other results. Kauhanen,
1,1
Koskela and Malý [12, Theorem A] show that functions u ∈ Wloc(Ω) whose weakn,1
partial derivatives belong to the Lorentz space L (Ω) have n-absolutely continuous
representatives in a domain Ω ⊂ Rn . Romanov [15, Theorem 2] generalizes thisresult to a locally s-Ahlfors regular metric space X for 1 ≤ p < s. He shows if u ishttps://doi.org/10.5186/aasfm.2019.4412
2010 Mathematics Subject Classification: Primary 46E35, 54E45.
Key words: Metric spaces, absolute continuity, Poincaré inequality, Sobolev spaces.

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a Poincaré Sobolev function [8] and the corresponding “Poincaré gradient” g belongs
to the Lorentz space Ls,1 (X), then u has an s-absolutely continuous representative.
In a compact and connected metric space X with finite 1-dimensional Hausdorff
measure, a classical result [1, Theorem 4.4.7] states that every two points can bejoined by the shortest curve. By curve, we mean a continuous mapping γ : [a, b] →X. The arc-length parametrization of this shortest curve is injective. We call thecurves without self intersections simple curves. The above results guarantees there
are sufficient simple curves in a compact, connected metric space X with finite 1dimensional Hausdorff measure. We define the absolutely continuous functions onsuch spaces as follows.
Definition 1.1. Let (X, d, H1 ) be a compact and connected metric measurespace with H1 (X) < ∞. A function u : X → R is absolutely continuous if for any
ǫ > 0, there is a positive number δ such thatX|u ◦ γi (ℓi ) − u ◦ γi (0)| < ǫ,ifor any countable collection of pairwise
P disjoint arc-length parametrized simple curves
γi : [0, ℓi ] → X with total length i ℓi < δ.

Remark 1.1. We denote the above class of absolutely continuous functions on
X by AC(X). Let X = [a, b], it is easy to verify this definition coincides with theclassical definition. It is also clear that the absolutely continuous functions defined
above are uniformly continuous when X is quasiconvex.
Remark 1.2. Let ui = u ◦ γi and Varℓ0i (ui ) denote the total variation of ui on
the interval [0, ℓi ] defined as
Varℓ0i (ui ) = sup

n−1
nX
k=1

o
|u ◦ γi (tk ) − u ◦ γi (tk+1 )| | 0 ≤ t1 < · · · < tn ≤ ℓi .

For simplicity, we write Var(ui ) = Varℓ0i (ui ). We can divide each simple curve In
Definition 1.1 into smaller
pieces and add the oscillation on the P
sub-curves. In thisPway, we can replace i |u ◦ γi (ℓi ) − u ◦ γi (0)| in the definition by i Var(ui ) and get
an equivalent characterization.

If u ∈ AC([a, b]), the pointwise derivative u′ exists almost everywhere and u′ ∈L ([a, b]). Likewise, if u ∈ AC(X), we prove that there exists an upper gradientassociated to u and this upper gradient belongs to L1 (X).
1

Theorem 1.1. Let (X, d, H1 ) be a compact and connected metric measure spacewith H1 (X) < ∞. If u ∈ AC(X), then there is an upper gradient g ∈ L1 (X) for u,that is, for any rectifiable curve γ : [a, b] → X, we haveˆ|u(γ(a)) − u(γ(b))| ≤ g.
γ

The construction of the upper gradient g can be roughly described as follows:
the one-dimensional space X in the assumption can be decomposed as a countablecollection of the images of pariwise disjoint simple curves γi : [0, ℓi ] → X and a null
set. On each simple curve piece, the definition that u ∈ AC(X) guarantees theabsolute continuity of u ◦ γi and the existence of the pointwise derivative (u ◦ γi )′ (t)for almost everywhere t ∈ [0, ℓi ]. Then the upper gradient g(x) can be defined as(u ◦ γi )′ (t) with t = γi−1 (x) up to a set of measure zero. To verify that this function g

Absolutely continuous functions on compact and connected 1-dimensional metric spaces283

is a upper gradient for u requires a careful comparison between the integral of g along
an arbitrary curve γ : [a, b] → X and a integral of the pointwise derivative (u ◦ γ0 )′over [0, ℓ0 ], where γ0 : [0, ℓ0 ] → X is the shortest curve connecting γ(a) and γ(b).
There are different approaches to extend the classical theory of Sobolev functions
to metric measure spaces [2, 6, 8, 17]. In this work, we mainly employ the definition
introduced by Shanmugalingam [17] and denote it by N 1,p (X). For an absolutely
continuous function u ∈ AC(X), if the upper gradient g associated to u (defined
explicitly in the proof of Theorem 1.1) is p-integrable, we write u ∈ AC p (X). By
definition, it follows that AC p (X) ∩ Lp (X) ⊂ N 1,p (X). On the other hand, we canverify that all Sobolev functions in N 1,p (X) belong to the class AC p (X) in our setting
X. Thus, we get the following result.
Theorem 1.2. Let 1 ≤ p < ∞ and (X, d, H1 ) be a compact, connected metricmeasure space with H1 (X) < ∞. Then u ∈ N 1,p (X) if and only if u ∈ AC p (X) andu ∈ Lp (X). In other words,
N 1,p (X) = AC p (X) ∩ Lp (X).
Remark 1.3. The upper gradient g we find in Theorem 1.1 is the least upper
gradient of u ∈ N 1,p (X), that is, if ρ ∈ Lp (X) is an upper gradient of u ∈ N 1,p (X),then g(x) ≤ ρ(x) holds almost everywhere. This follows from the construction of g
described above and Lemma 3.1. More details can be found in the proof of Theorem 1.2.
The notion of an abstract Poincaré inequality on metric measure spaces wasintroduced by Heinonen and Koskela [10]. Metric measure spaces that are doubling and support an abstract Poincaré inequality provide a good structure to study
the first-order analysis. In some spaces, a pure geometric characterization of the
Poincaré condition is possible. For example, Durand-Cartagena, Jaramillo and Shanmugalingam [3] show a connected complete doubling metric measure space supports
a ∞-Poincaré inequality if and only if it is thick quasiconvex. In Q-Ahlfors regularspaces with Q > 1, they also obtain a characterization of p-Poincaré condition forp > Q − 1 in terms of a quantitative estimate of the p-modulus of the family of
all quasiconvex curves [4]. In our settings, we prove that for each compact and 1Ahlfors regular space, supporting a p-Poincaré inequality for 1 ≤ p ≤ ∞ is equivalent
to being quasiconvex.
Theorem 1.3. Let (X, d, H1 ) be a compact 1-Ahlfors regular metric measurespace. Then it supports p-Poincaré inequality for 1 ≤ p ≤ ∞ if and only if X is
quasiconvex.
Let X be a complete metric space equipped with a doubling measure and 1 ≤q < p. If X supports a q-Poincaré inequality, the Sobolev spaces N 1,p (X) defined byupper gradients, the Sobolev spaces P 1,p (X) defined by Poincaré inequalities, and the
Sobolev spaces M 1,p (X) defined by a pointwise inequality using maximal functions
all coincide [7, Theorem 11.3]. Consequently, we obtain the following corollary.
Corollary 1.4. Let (X, d, H1 ) be a compact, quasiconvex, 1-Ahlfors regular metric measure space and 1 < p < ∞. Then
AC p (X) ∩ Lp (X) = N 1,p (X) = P 1,p (X) = M 1,p (X).
This paper is organized in the following way. In Section 2, we review several
basic definitions and some well-known results. In Section 3, we prove Theorem 1.1
and Theorem 1.2. In Section 4, we prove Theorem 1.3 and Corollary 1.4 follows.

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Acknowledgments. The author thanks her advisor Piotr Hajłasz for bringing
this topic to her attention and many helpful discussions. She also thanks Nageswari
Shanmugalingam and Changyu Guo for their interests and comments. The author
is grateful to the anonymous referees for several comments leading to improving theexposition of the paper.
2. Definitions and notations
In this section, we will give some basic definitions and notations, as well as a listof several known results. Most of the results in this section can be found in [11].
2.1. s-Ahlfors regular space and doubling space.
Definition 2.1. We say that a space (X, d, µ) is s-Ahlfors regular if there is a
fixed constant C, such thatC −1 r s ≤ µ(B(x, r)) ≤ Cr s ,where x ∈ X and 0 < r < diam(X).
If X is s-Ahlfors regular with respect to µ, we can replace µ by the Hausdorff
measure Hs without losing essential information [9, Exercise 8.11].
We say that µ is a doubling measure if there is a fixed constant C > 0 such thatµ(B(x, 2r)) ≤ Cµ(B(x, r)) for each x ∈ X and all r > 0. If X is equipped with adoubling measure µ, we call X is a doubling space.
2.2. Upper gradients and Newtonian Sobolev spaces N 1,p (X). Rectifiable curve always admits an arc-length parametrization. Let γ : [a, b] → X be a
rectifiable curve and g : γ([a, b]) → [0, ∞] be a Borel measure function, we define
ˆ
ˆ ℓ
g :=
g(γ̃(t)) dt,
γ0

where γ̃ : [0, ℓ] → X is the arc-length parametrization of γ.
Definition 2.2. Let u : X → R be a Borel function. We say that a Borel
function g : X → [0, ∞] is an upper gradient of u ifˆ|u(γ(a)) − u(γ(b))| ≤ gγfor every rectifiable curve γ : [a, b] → X.
e 1,p (X, d, µ) is the class of all Lp -integrable functions on X
Let 1 ≤ p < ∞, N
e 1,p (X, d, µ) we
for which there exists a p-integrable upper gradient. For each u ∈ N
associate a seminorm
kukNe 1,p (X,d,µ) = kukLp + inf kgkLp ,gwhere the infimum is taken over all upper gradients of u.
e 1,p (X, d, µ) by u ∼ v if
Definition 2.3. We define an equivalence relation in N
1,pku − vkNe 1,p (X,d,µ) = 0. Then the space N (X, d, µ) is defined as the quotient space
e 1,p (X, d, µ)/ ∼ and it is a Banach space equipped with the norm
NkukN 1,p = kukNe 1,p .

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285

2.3. Poincaré Sobolev spaces P 1,p (X) and Hajłasz Sobolev spaces
M 1,p (X).
Definition 2.4. Fix σ ≥ 1 and 0 < p < ∞. We say that a pair (u, g), u ∈ L1 (X),0 ≤ g ∈ Lp (X) satisfies the p-Poincaré inequality if the following inequality holds: p1p|u − uB |dµ ≤ r
g dµ
B

σB

on every ball B of radius r and σB ⊂ X.
In this paper, we call g in the above inequality the “Poincaré gradient” for u.
The class of u ∈ Lp (X) for which there exists 0 ≤ g ∈ Lp (X) so that the pair (u, g)satisfies the p-Poincaré inequality will be denoted by Pσ1,p (X) and[P 1,p (X) =Pσ1,p (X).
σ≥1

Definition 2.5. For 0 < p < ∞ we define M 1,p (X) to be the set of all functionsu ∈ Lp (X) for which there exists g ≥ 0 such that|u(x) − u(y)| ≤ d(x, y)(g(x) + g(y)) µ-a.e.
Denote by D(u) the class of all nonnegative Borel functions g that satisfy the above
inequality. Thus u ∈ M 1,p (X) if and only if u ∈ Lp (X) and D(u) ∩ Lp (X) 6= ∅. The
space M 1,p (X) is linear and we equip it with the norm||u||M 1,p = ||u||Lp + inf ||g||Lp ,g∈D(u)

for 1 ≤ p < ∞.
2.4. Spaces supporting a Poincaré´ inequality. Recall that uB stands for1
an integral average, that is, uB = µ(B)u dµ. A metric measure space (X, d, µ)Bsupporting Poincaré inequality is defined in the following way.
Definition 2.6. Let p ≥ 1. A metric measure space (X, d, µ) is said to support
a p-Poincaré inequality if there exists constants C > 0 and λ ≥ 1 such that for all
measurable functions, the following holds for every pair of functions u : X → R andg → [0, ∞] where u is measurable, and g is an upper gradient for u: p1p|u − uB | dµ ≤ C diam B
g dµ ,B

λB

on every ball B.

A metric space X is quasiconvex if every two points can be joined by a curvewith length comparable to the distance of these two points. If X is complete, doubling and supports a p-Poincaré inequality for p ≥ 1, then X is quasiconvex [2,Theorem 17.1.][8, Proposition 4.4].
The following theorem can be found in [7, Theorem 11.3].
Theorem 2.1. Let (X, d, µ) be a complete metric space equipped with a doubling measure. If 1 < p < ∞ and the space supports the q-Poincaré inequality forsome 1 ≤ q < p, thenN 1,p (X) = P 1,p (X) = M 1,p (X).
2.5. Area Formula. We need the following generalization of the Euclidean
Area Formula to the case of Lipschitz maps f from the Euclidean space Rn into a
metric space X. The proof can be found in [13, Corollary 8].

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Theorem 2.2. Let f : Rn → X be Lipschitz. ThenˆˆX
g(x)Jn (mdfx ) dx =g(x) dHn (y)Rn

X x∈f −1 (y)for any Borel function g : Rn → [0, ∞] andˆˆg(f (x))Jn (mdfx ) dx =g(y)H0(A ∩ f −1 (y)) dHn (y)AX

for A ⊂ Rn is measurable and any Borel function g : X → [0, ∞].
We apply the above theorem to an injective, arc-length parametrized curve. Let
f = γ and γ : [0, ℓ] → X. In this case, J1 (mdfx ) equals to the metric derivative
defined asd(γ(t + h), γ(t))|γ̇|(t) = lim
,
h→0
|h|
and |γ̇|(t) = 1 almost everywhere for t ∈ [0, ℓ]. Let Γ = γ([0, ℓ]) and g : X → [0, ∞]be a Borel function. It follows from Theorem 2.2 thatˆ ℓ
ˆg(γ(s)) ds = g(y) dH1(y).
0ΓThis implies that H1 (Γ) = ℓ for an injective, arc-length parametrized curve. If
N ⊂ X and H1 (N) = 0, the above Area Formula also shows that H1 (γ −1 (N)) = 0.
2.6. Rectifiability of 1-dimensional connected spaces. We list several
important results about the parametrization of compact and connected 1-dimensional
metric spaces. The first result is proved by Schul [16, Lemma 2.3] and gives a global
Lipschitz parametrization of finite 1-dimensional compact, connected metric spaces.
Lemma 2.3. Let K ⊂ X be a compact connected set of finite H1 measure.
Then there is a Lipschitz function γ : [0, 1] → K such that Image(γ) = K andkγkLip ≤ 32H1 (K). Moreover, if K is 1-Ahlfors-regular, thenR
≤ H1 (γ −1 (B(x, R))) ≤ CR ∀x ∈ K, 0 < R ≤ diam(K),C
where C is a constant depending only on the 1-Ahlfors-regularity constant of the set
K.
The proofs of the following two classical results can be found in [1, Theorem 4.4.7,
Theorem 4.4.8].
Theorem 2.4. (First Rectifiability Theorem) If E is complete and C ⊂ E is a
closed connected set such that H1 (C) < ∞, then C is compact and connected byrectifiable curves.
Actually, we can replace the rectifiable curve joining any two points x, y by geodesic, the shortest curve connecting x, y in C in the above theorem [7, Theorem 3.9].
Theorem 2.5. (Second Rectifiability Theorem) If E is complete, C ⊂ E isclosed and connected, and H1 (C) < ∞, then there exist countably many arc-lengthparametrized simple curves γi : [0, ℓi ] → C such that1

H (C \∞[i=1

γi ([0, ℓi ])) = 0.

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287

We briefly describe the construction of this parametrization. Since C is compact,
we can choose x, y ∈ C such thatd(x, y) = diam (C).
By First Rectifiability Theorem (Theorem 2.4), we can join x, y by an arc-lengthparametrized geodesic γ0 : [0, ℓ0 ] → C and we denote the range of this curve as Γ0 .
Suppose that we have already constructed Γ0 , · · · , Γk with the following properties:(1) Γi ⊂ C, i = 0, · · · , k;(3) Each curve γi : [0, ℓi ] → C with Γi = γi ([0, ℓi ]) is an arc-length parametrized
geodesic;
S(2) Each intersection Γi ∩ j<i Γj consists of a single point, for i = 1, · · · , k.
Let

dk = sup d x,
x∈C

k
[

i=0

Sk


Γi .

If dk = 0, then C = i=0 Γi and we are done. If dk > 0 for all k, by compactness
we can choose xk ∈ C and yk ∈ ∪ki=0 Γi such that d(xk , yk ) = dk . Connect xk
and yk with an arc-length parametrized geodesic γk+1 such that γk+1(0) = xk andγk+1(ℓk+1 ) = yk+1. Let
)
(
k
[
Γi
t̃ = inf t ∈ [0, ℓk+1] | γk+1(t) ∈i=0

and define Γk+1 = γk+1 ([0, t̃]). We get that Γk+1 ⊂ X is an arc-length parametrizedS
geodesic and the intersection of Γk+1 and Ski=0 Γi consists of one single point γk+1 (t̃).
We can continue this construction. Since ∞i=0 Γi may not be closed, we may haveX\∞[

Γi 6= ∅.

i=0

We omit the proof for H1 (X \S∞

i=0 Γi ) = 0.

Remark 2.1. From the above construction, if two curves intersect with eachother, the intersection point must be the endpoint of one of them. Thus, if weremove the endpoints of these curves, they are pairwise disjoint.
Remark 2.2. Since we choose γi to be geodesic in each step, for any two points
x, y ∈ Γi , γi is the shortest curve joining x and y in C.
3. Characterization of Sobolev functions by absolute continuity
3.1. Absolutely continuous functions belong to N 1,1 (X).
Proof of Theorem 1.1. By Second Rectifiability Theorem (Theorem 2.5), we
know that there is a countable collection of simple curves in X and a set N1 with
H1 (N1 ) = 0 such that∞[X=
γi ((0, ℓi )) ∪ N1 .
i=1

We denote the range of γi by Γi = γi (0, ℓi ) and denote u ◦ γi by ui . The intersectionof Γi and Γj is empty for i 6= j. If u ∈ AC(X), then ui ∈ AC((0, ℓi )). Thus, u′i (t)Xiaodan Zhou

288

exists almost everywhere for t ∈ (0, ℓi ). Moreover, we haveˆ ℓiui (ℓi ) − ui (0) =u′i (s) ds,
0

and

ˆ ℓi

Var(ui ) =|u′i (s)| ds.
S∞0
We denote the collection of points x ∈ i=1 Γi such that u′i (γi−1 (x)) does not exist as
N2 , and let N0 = N1 ∪ N2 . It is clear that H1 (N0 ) = 0. Then we define a function
g : X → R as follows,(|u′i (γi−1 (x))| if x ∈ X \ N0 ,(1)g(x) =∞
if x ∈ N0 .
This function g is integrable on X. In fact, by the decomposition of X, we have
ˆ
ˆ
Xˆ
1
1
g dH1 .
g dH =
g dH =
X

X\N1

Γi

i

Apply Area Formula to each arc-length parametrized simple curve Γi , it follows
XˆX ˆ ℓi1g dH =g(γi (s)) ds.
Γi

i

0

i

Since the L1 (γi−1 (N2 )) = 0 and by definition of the function g, we getˆX ˆ ℓiX ˆ ℓiX
1
g dH =g(γi (s)) ds =|u′i (s)| ds =Var(ui ).
X

0

i

i

0

i

1

Since γi is injective and H (X) < ∞, it implies thatX
X
H1 (Γi ) ≤ H1 (X) < ∞.
ℓγi =
i

i

P
For any ǫ > 0, there exists aPnatural number n0 such that ∞i=n0 +1 ℓγi < ǫ. By
Definition 1.1, it implies that ∞
Var(u
)
can
be
sufficiently
small. Thus,i
i=n0 +1
ˆ
n0
∞
X
X
X
1
g dH =
Var(ui ) =Var(ui ) +Var(ui ) < ∞.
Xi

i=n0 +1i=1

We next prove that g is an upper gradient for the function u ∈ AC(X), that is,for any rectifiable curve γ : [a, b] → X, we haveˆ|u(γ(a)) − u(γ(b))| ≤ g ds.
γ

Let γ : [a, b] → X be a rectifiable curve and Γ = γ([a, b]). Since Γ is compact,
connected and H1 (Γ) < ∞, there exists a shortest curve joining γ(a) and γ(b) in Γ.
We denote the arc-length parametrization of this injective curve by γ0 : [0, ℓ] → Γ
with
(2)

γ0 (0) = γ(a) and γ0 (ℓ) = γ(b).

Let Γ0 = γ0 ([0, ℓ]). It follows from Area Formula that
ˆ ℓ
ˆ
ˆ
ˆ
ˆ
1
1
(3)
g ds =
g(x) dH ≤ g(x) dH ≤ g ds.
g(γ0(s)) ds =
γ0

0

Γ0

Γ

γ

Absolutely continuous functions on compact and connected 1-dimensional metric spaces

289

Since u ∈ AC(X), it follows that u0 = u ◦ γ0 ∈ AC([0, ℓ]) and u′0 exists
almostS
1everywhere. Let I = {i ∈ N : H (Γ0 ∩Γi ) 6= 0}. Then Γ0 is the union of i∈I (Γ0 ∩Γi )and a null set.
Let i ∈ I and x ∈ Γi ∩ Γ0 such that u′0 (γ0−1 (x)), u′i (γi−1 (x)) both exist and
t = γ0−1 (x) is a density point of γ0−1 (Γi ∩ Γ0 ). By definition,|u0(t + h) − u0 (t)||u(γ0(t + h)) − u(γ0(t))|
= lim
.
h→0
h→0
|h|
|h|

|u′0(t)| = lim

Since t is a density point in the measurable set γ0−1 (Γi ∩ Γ0 ), we can take a sequence−1of nonzero number {hk }∞k=1 such that t + hk ∈ γ0 (Γi ∩ Γ0 ) and limk→∞ hk = 0. We
denote γi (s) = γ0 (t) and γi (sk ) = γ0 (t + hk ). By Remark 2.2, the curve γi is theshortest curve joining γi (s) and γi (sk ). Since γ0 and γi are both parametrized by
arc-length, it follows that|s − sk | = ℓγi (s,sk ) ≤ ℓγ0 (t,t+hk ) = |hk |.
This implies that|u(γ0(t + hk )) − u(γ0(t))|
k→∞
|hk |
|u(γi (sk )) − u(γi (s))|= |u′i (s)|.
≤ lim
k→∞
|sk − s||u′0 (t)| = limThus,|u′0(t)| ≤ |u′i (s)| = |u′i (γi−1 (γ0 (t)))| = g(γ0(t))
holds almost everywhere for t ∈ [0, ℓ]. In fact, Area Formula guarantees H1 (γ0−1 (N)) =H1 (N) = 0 for any null set N ⊂ X. It follows that
ˆ ℓ
ˆ ℓ
ˆ
′
(4)
|u0(ℓ) − u0 (0)| ≤|u0 (t)| dt ≤g ds.
g(γ0(t)) dt =
0

0

γ0

Combining inequalities (2), (3), (4), we get|u(γ(a)) − u(γ(b))| = |u(γ0(0)) − u(γ0 (ℓ))| = |u0 (0) − u0 (ℓ)| ≤ˆγ0

g ds ≤

ˆ

g ds.

γ

Thus, we verify that g ∈ L1 (X) is an upper gradient of u and the proof is complete.

3.2. Sobolev functions in N 1,p are absolutely continuous. To conclude
the proof of Theorem 1.2, we need the following lemma [11, Proposition 6.3.3].
Lemma 3.1. Let u : X → R be a function and γ : [0, ℓ] → X be an arc-lengthparametrized rectifiable curve in X. Assume that ρ : X → [0, ∞] is a Borel function
such that ρ is integrable on γ and the pair (u, ρ) satisfies the upper gradient inequality
on γ and each of its compact subcurves. Then u ◦ γ is absolutely continuous and theinequality|(u ◦ γ)′ (t)| ≤ (ρ ◦ γ)(t),holds for almost every t ∈ [0, ℓ].
Proof of Theorem 1.2. If u ∈ AC(X) ∩ L1 (X), then Theorem 1.1 implies thatu ∈ N 1,1 (X). If we further assume that u ∈ AC p (X), that is, the upper gradient g
defined in (1) belongs to Lp (X). It follows that u ∈ N 1,p (X) andAC p (X) ∩ Lp (X) ⊂ N 1,p (X).

Xiaodan Zhou

290

On the other hand, if u ∈ N 1,p (X), and ρ is a p-integrable upper gradient of u. For
any arc-length parametrized simple rectifiable curves γ : [0, ℓ] → X, ρ is integrable
along γ and H1 (Γ) = ℓ by Area Formula. Since (u, ρ) satisfies the following upper
gradient inequalityˆ ℓˆ
|u(γ(ℓ)) − u(γ(0))| ≤ρ(γ(s)) ds = ρ(y) dH1(y),0Γthe absolute continuity of integral implies immediately that u ∈ AC(X).
Moreover, the upper gradient g defined in (1) is the least upper gradient foru ∈ N 1,p (X). Let γi be the simple curves constructed in the Second Rectifiability
Theorem. Lemma 3.1 implies that|u′i (t)| = |(u ◦ γi )′ (t)| ≤ (ρ ◦ γi )(t).
For almost every point x ∈ X, the above inequality implies thatg(x) = |u′i (γi−1 (x))| ≤ ρ(x).
The upper gradient g defined in (1) is bounded by a p-integrable function ρ almost
everywhere in X. It implies that g ∈ Lp (X) and u ∈ AC p (X). Thus, we get
N 1,p (X) ⊂ AC p (X) ∩ Lp (X).
The proof of Theorem 1.2 is complete.

Note the proofs of the above two theorems also work in the case p = ∞.
4. Space supporting Poincaré inequality
Proof of Theorem 1.3. Since every complete and doubling metric measure spacethat supports a p-Poincaré inequality for p ≥ 1 is quasiconvex [8, Proposition 4.4]
[11, Theorem 8.3.2], the necessity part of the theorem is clear.
On the other hand, by Hölder’s inequality, when the compact 1-Ahlfors regularmetric space is quasiconvex, it suffices to show that X supports a 1-Poincaré inequality. Let B(O, r) ⊂ X be an arbitrary ball in X and x, y ∈ B. There exists a
shortest rectifiable curve connecting x and y [7, Theorem 3.9]. We denote the arclength parametrization of this shortest curve by γ : [0, ℓ] → X such that γ(0) = x
and γ(ℓ) = y. At the same time, X being quasiconvex guarantees that there is a
curve γ̃(x, y) joining x and y with length ℓγ̃(x,y) such thatℓ ≤ ℓγ̃(x,y) ≤ Cd(x, y).
Let u be a Borel function and g be an upper gradient of u. Thenˆ ℓ(5)|u(x) − u(y)| ≤g(γ(s)) ds.
0

Denote Γ = γ([0, ℓ]) and λ = 3C, where C is the quasiconvexity constant of X. We
next verify that the whole curve Γ ⊂ B(O, λr). By triangle inequality, d(z, O) ≤d(z, x) + d(x, O). Let ℓγ(x,z) denote the length of the shortest curve joining x and z.
We have d(z, x) ≤ ℓγ(x,z) ≤ ℓ. Thus,d(z, O) ≤ ℓ + d(x, O) ≤ ℓγ̃(x,y) + r ≤ λr.
The fact that Γ ⊂ B(O, λr) implies that
ˆ ℓ
ˆ
ˆ
1
(6)
g(γ(s)) ds = g(y) dH ≤
0

Γ

λB

g(y) dH1.

Absolutely continuous functions on compact and connected 1-dimensional metric spaces

291

Finally, combining (5) and (6), we get1ˆ ℓ|u(x) − u(y)| dH ≤g(γ(s)) ds dH1
B
B 0
ˆ
≤
g(x) dH1 dH1 = H1 (λB)
g dH1 .

|u(x) − uB | ≤
(7)

B

λB

λB

Since X is 1-Ahlfors regular, it follows that|u(x) − uB | dH1 ≤ C diam(B)B

g dH1 .



λB

Finally, combining Theorem 1.2, Theorem 1.3 and Theorem 2.1, we get Corollary 1.4.
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Received 5 January 2018 • Accepted 13 August 2018