The Space of Functions of Bounded Variation on Curves in Metric Measure Spaces

Home , Borel measure, Bounded variation

CONFORMAL GEOMETRY AND DYNAMICS An Electronic Journal of the American Mathematical Society Volume 20, Pages 81–96 (April 19, 2016) http://dx.doi.org/10.1090/ecgd/291

THE SPACE OF FUNCTIONS OF BOUNDED VARIATION ON CURVES IN METRIC MEASURE SPACES

O. MARTIO Dedicated to Jan Mal´yonhis60th birthday

Abstract. The approximation modulus, AM–modulus for short, was deﬁned in an earlier paper by the author. In this paper it is shown that an L1(X)– function u in a metric measure space (X, d, ν) can be deﬁned to be of bounded variation on AM–a.e. curve in X without an approximation of u by Lipschitz or Newtonian functions. The essential variation of u on AM almost every curve is bounded by a sequence of non-negative Borel functions in L1(X). The space of such functions is a Banach space and there is a Borel measure associated with u.ForX = Rn this gives the classical space of BV functions.

1. Introduction Since the pioneering work of Ahlfors and Beurling the modulus method has been widely used in conformal geometry in Rn, n ≥ 2, and, more recently, in metric measure spaces; see [10, 11]. Fuglede [8] showed that the method is useful in the theory of function spaces and in the study of regularity properties of functions in these spaces. The modulus method has also been applied to function spaces in metric measure spaces (see [4,5,15]), where it has a more fundamental role than in Rn. In this paper a new modulus, the AM–modulus, is used to construct a function space of functions of bounded variation (BV) in a metric measure space X.The construction corresponds to the construction of the so-called Newtonian spaces N 1,p(X)inX; see [5, 15]. Instead of the concept of a (weak) upper gradient we use the concept of a BV upper bound which consists of a sequence of non-negative Borel functions in L1(X). An N 1,p(X) function need not be absolutely continuous on every curve; it is necessary to omit the family of curves of Mp–modulus zero. Similarly a BV function need not be of bounded variation on every curve. This is already evident in R where the function u = χ{0} is not BV on every curve. For example, u is not BV on the curve which crosses 0 infinitely often. The AM–modulus takes care of this situation. It is customary to consider BV functions in X defined only almost everywhere. From the approximation point of view L1(X) is a good choice for the base space and this approach is used both in Rn (see [7]) and in X (see [14]). In [1,2,6] several alternative definitions for BV functions in X based on approximation by Lipschitz functions are discussed. As in the case of the Newtonian spaces our approach does not use approximation by smooth functions. It turns out, as in the Newtonian

Received by the editors September 24, 2015 and, in revised form, February 10, 2016. 2010 Mathematics Subject Classiﬁcation. Primary 26A45; Secondary 26B30, 30L99. Key words and phrases. Metric measure space, AM–modulus, functions of bounded variation.

c 2016 American Mathematical Society 81 82 O. MARTIO space case, that in Rn our approach and the approximation approach lead to the same space. The AM–modulus can be applied to other base spaces than L1(X) (see [13, Section 4]), but we consider L1(X) since it naturally leads to a Banach space of BV functions in X.ThespaceL1(X) also has the advantage that its functions are well defined on M1–a.e. curve and thus on AM–a.e. curve. In a forthcoming paper with V. Honzlová–Exernova and J. Malý we show that n there exist curve families Γ in R , n ≥ 1, such that M1(Γ) = ∞ but AM(Γ) = 0. Since AM(Γ) ≤ M1(Γ) for every curve family Γ in X, this shows that the AM– modulus is essentially weaker than the M1–modulus. After preliminaries in Section 2 we prove the Banach property of the space BVAM (X) of BV functions in Section 3. Section 4 is devoted to the construction of a Borel measure associated with u. The construction resembles the construction due to Miranda in [14] who used approximation by Lipschitz functions with L1 bounded upper gradients for u. InSection5thecaseX = Rn is considered. We use minimal assumptions on the space (X, d)andthemeasureν. Our notation is standard.

2. Preliminaries We recall the basic properties of functions of bounded variation and the AM– modulus and introduce the concept of a BVAM upper bound for a function u ∈ L1(X). Let (X, d, ν) be a metric measure space where d is a metric and ν is a regular Borel measure in X, We use the deﬁnition in [5, I.1.1] for regularity and, in particular, ν is an outer measure in X. In addition we assume that ν satisﬁes

(2.1) 0 <ν(B(x, r)) < ∞ for each ball B(x, r)inX. In Section 4 we make some additional assumptions on the space (X, d). Functions of bounded variation. Suppose that u :[a, b] → R ∪{∞, −∞} and there is a set C ⊂ [a, b] such that m1([a, b] \ C)=0andu|C is of bounded variation on C. The latter means that n V (u, C)=sup |u(xi) − u(xi−1)| < ∞ i=1 where the supremum is taken over all ﬁnite sequences x0

(2.2) V (û, [a, b]) = V (û, C) THE SPACE OF FUNCTIONS ON CURVES IN METRIC MEASURE SPACES 83 for every C ⊂ [a, b]offullmeasure,sinceifV (û, [a, b]) >V(û, C), then by [13, Lemma 2.1] there is a function v in [a, b] such that v =û in C and V (v, [a, b]) = V (v, C)=V (û, C) so thatu ˆ is not of minimal variation.

1 Lemma 2.1. Suppose that ui → u in L ([a, b]).Then

(2.3) V (û, [a, b]) ≤ lim inf V (ûi, [a, b]). i→∞ → 1 Proof. Note thatu î uˆ in L ([a, b]). Choose a subsequence (ûij )of(ûi) such that

lim V (ûi , [a, b]) = lim inf V (ûi, [a, b]) j→∞ j i→∞ → and then a subsequence of (ûij ), denoted again by (ûij ), such thatu îj uˆ in C where m1([a, b] \ C) = 0. From (2.2) we obtain

V (û, [a, b]) = V (û, C) ≤ lim inf V (ûi ,C) j→∞ j

= lim inf V (ˆui , [a, b]) = lim inf V (ˆui, [a, b]) j→∞ j i→∞ as required.

M1–andAM–modulus. AcurveinX is a continuous mapping γ :[a, b] → X. We can always assume that a curve is non-constant and rectifiable and then we parametrize γ by arc length, i.e. γ :[0,l(γ)] → X where l(γ) is the length of γ. This is due to the fact that both the M1–andtheAM–modulus of the family of all non-rectifiable curves in X is zero. Note that it is essential that we consider curves which are defined on compact intervals. The M1–andAM–modulus can be defined on families of curves which are defined on open or half open intervals and which are only locally rectifiable. Then it turns out that there are families of non-rectifiable curves whose M1–andtheAM–modulus are not zero. Let Γ be a family of curves in X. We first recall the concept of Mp–modulus, p ≥ 1. A non–negative Borel function ρ is M –admissible for Γ if p ρds≥ 1 γ for every γ ∈ Γ. The M –modulus of Γ is defined as p p Mp(Γ) = inf ρ dν X where the infimum is taken over all admissible functions ρ. For the theory of Mp– modulus see [5, 8]. In this paper we only use the M1–modulus. A sequence of non-negative Borel functions ρi, i =1, 2,..., is AM–admissible for Γ if (2.4) lim inf ρ ds ≥ 1 →∞ i i γ for every γ ∈ Γ. The approximation modulus, AM–modulus for short, of Γ is defined as (2.5) AM(Γ) = inf {lim inf ρ dν} →∞ i (ρi) i X where the infimum is taken over all AM–admissibe sequences (ρi)forΓ. 84 O. MARTIO

We say that a property holds on Mp–almost every (a.e.) curve or on AM–a.e. curve if it holds for every curve except for a family Γ with Mp(Γ) = 0 or AM(Γ) = 0, respectively. Since AM(Γ) ≤ M1(Γ) for each curve family Γ in X, the property which holds on M1–a.e. curve holds on AM–a.e. curve as well. For the properties of the AM–modulus see [13]. BVAM upper bounds. Let γ be a curve in X and u a function such that u◦γ is a.e. defined in [0,l(γ)]. If V (u◦ γ,[0,l(γ)]) < ∞,wesetV (u, γ)=V (u◦ γ,[0,l(γ)]) and if there is no set C ⊂ [0,l(γ)] such that m1([0,l(γ)] \ C)=0andV (u ◦ γ,C) < ∞, then we set V (u, γ)=∞. Suppose that u ∈ L1(X). By [5, Lemma 1.43 and Proposition 1.37] the integral uds γ is well defined and finite on M1–a.e. curve γ. In particular the function u ◦ γ is a.e. defined on [0,l(γ)] and hence V (u, γ) is defined on M1– and hence on AM–a.e. curve γ in X. A sequence (gi) of non-negative Borel functions is called a BVAM upper bound for u ∈ L1(X)if (2.6) V (u, γ) ≤ lim inf g ds →∞ i i γ on AM–a.e. curve γ in X.Notethatif(gi)isaBVAM upper bound for u,then every subsequence of (gi)isalsoaBVAM upper bound for u. The next lemma gives alternative characterizations for (2.6).

1 Lemma 2.2. Let u ∈ L (X) and (gi) a sequence of non-negative Borel functions in X. Then the following conditions are equivalent:

(2.7) (gi) is a BVAM upper bound for u,

(2.8) V (u, γ|[t1,t2]) ≤ lim inf gi ds for all 0 ≤ t1

(2.9) |u(γ(t2)) − u(γ(t1))|≤lim inf gi ds for a.e. t1,t2 ∈ [0,l(γ)], i→∞ γ|[t1,t2] where (2.8) and (2.9) are supposed to hold on AM–a.e. curve γ in X. Proof. Suppose that (2.7) holds. Since (2.6) holds on every curve γ, it holds on every subcurve of γ as well and so (2.8) follows. Clearly (2.8) implies (2.9) and it remains to show (2.9) ⇒ (2.7). Suppose that (2.9) holds on a curve γ.LetC ⊂ [0,l(γ] be a set of full measure and (2.9) holds for all t1,t2 ∈ C, t1

n−1 ≤ lim inf gi ds ≤ lim inf gi ds i→∞ | i→∞ j=1 γ [aj ,aj+1] γ and this shows that (gi)isaBVAM upper bound for u. The lemma follows. THE SPACE OF FUNCTIONS ON CURVES IN METRIC MEASURE SPACES 85

→ 1 Lemma 2.3. If ui u in L (X), then the sequence (ui) has a subsequence (uij ) such that

V (u, γ) ≤ lim inf V (ui ,γ) j→∞ j for AM–a.e. curve γ in X.

Proof. By the Fuglede lemma [5, Lemma 2.1] the sequence (ui) has a subsequence (u ) such that ij | − | → uij u ds 0 γ as j →∞for M1–a.e., and thus for AM–a.e., curve γ in X. The conclusion now follows from Lemma 2.1.

If (gi)and(hi)areBVAM upper bounds of u in X,then(max(gi,hi)) is clearly a BVAM upper bound for u but, as easy examples show, (min(gi,hi)) is not. In the following lemma we consider the situation where the BVAM upper bounds are deﬁned on diﬀerent open sets.

j Lemma 2.4. Let Ωj , j =1, 2,beopensetsinX and (gi ) a BVAM upper bound 1 for a function u ∈ L (Ω1 ∪ Ω2) in Ωj, j =1, 2. Then the sequence (gi), ⎧ 1 ∈ \ ⎨ gi (x),x Ω1 Ω2, 1 2 ∈ ∩ gi(x)=⎩ max(gi (x),gi (x)),x Ω1 Ω2, 2 ∈ \ gi (x),x Ω2 Ω1, is a BVAM upper bound for u in Ω1 ∪ Ω2.

Proof. Note that (gi)isaBVAM upper bound for u both in Ω1 and Ω2 and hence (2.10) V (u, γ) ≤ lim inf g ds →∞ i i γ whenever γ is a subcurve of γ with γ ⊂ Ωj , j = 1 or 2. Fix a curve γ in Ω1 ∪ Ω2.Sinceγ([0,l(γ)]) is compact in Ω1 ∪ Ω2,thereisδ>0 such that γ lies in Ω1 or in Ω2 whenever γ is a subcurve of γ with l(γ ) <δ.By (2.10) and Lemma 2.2 there is a set C ⊂ [0,l(γ)] with m1([0,l(γ)] \ C)=0and

(2.11) |u(γ(t2)) − u(γ(t1))|≤lim inf gi ds i→∞ γ|[t1,t2] whenever t1,t2 ∈ C,0

n ≤ lim inf gi ds ≤ lim inf gi ds i→∞ | i→∞ k=1 γ [tk,tk−1] γ and the lemma follows. 86 O. MARTIO

3. The space BVAM (X)

The space, abreviated as BVAM (X), of the functions of bounded variation on 1 AM–a.e. curve in X consists of all functions u ∈ L (X) which have a BVAM upper bound (g ) such that i (3.1) lim inf g dν < ∞. →∞ i i X We call u BV (X)= u L1(X) + DBV u (X). where D u (X) = inf{lim inf g dν :(g )isaBV upper bound for u} BV →∞ i i AM i X the BVAM norm of u in X. The essential implication of (3.1) is the following; see [13]. For completeness we give a proof.

1 Lemma 3.1. Suppose that u ∈ L (X) has a BVAM upper bound (gi) in X as in (3.1).Then (3.2) V (u, γ) ≤ lim inf g ds < ∞ →∞ i i γ for AM–a.e. curve γ in X. Proof. For each k =1, 2,... let Γ be the family of all curves γ in X such that k lim inf g ds ≥ k. →∞ i i γ Then the sequence (g /k)isAM–admissible for Γ and hence i k g M AM(Γ ) ≤ lim inf i dν = k →∞ i X k k where M = lim inf g dν. →∞ i i X Since Γ∞ = {γ : lim inf g ds = ∞} →∞ i i γ satisﬁes Γ∞ ⊂ Γk for each k, it follows that AM(Γ∞) = 0 and thus (3.2) holds as required.

The following property of a function u ∈ BVAM (X) is often useful. For each ε>0 there is a BV upper bound (g )foru in X such that AM i

(3.3) gi dν ≤ DBV u (X)+ε X for all i and (3.2) holds for AM–a.e. curve γ in X. Remark 3.2. A locally Lipschitz function u ∈ L1(X) with an upper gradient also in 1 1,1 L (X) and a function u in the Newtonian space N (X)isinthespaceBVAM (X) since for such a function u the sequence (gi), where gi = g and g is an upper gradient or a 1–weak upper gradient of u, respectively, is a BVAM upper bound for u. For the concept of an upper gradient and a 1–weak upper gradient see [4,5,6,15]. THE SPACE OF FUNCTIONS ON CURVES IN METRIC MEASURE SPACES 87

1 Lemma 3.3. The space BVAM (X) is a linear subspace of L (X) and u BV (X) is a norm in BVAM (X).

Proof. If u ∈ BVAM (X)andλ ∈ R, then obviously λu ∈ BVAM (X)and 1 λu BV (X) = |λ| u BV (X).Ifu =0inL (X), then u ◦ γ = 0 a.e. in [0,l(γ)] on M1-a.e. curve γ in X; see Section 2. Hence (gi),gi =0,i=1, 2,..., is a BVAM upper bound for u and thus u BV (X)=0. It remains to show that u + v ∈ BVAM (X) provided that u, v ∈ BVAM (X)and that the triangle inequality holds. Let u, v ∈ BVAM (X)andlet(gi)and(hi)be BVAM upper bounds for u and v, respectively. Now (gi + hi)isaBVAM upper bound for u + v. Indeed, for AM–a.e. curve γ in X

V (u + v, γ) ≤ V (u, γ)+V (v, γ) ≤ lim inf g ds + lim inf h ds ≤ lim inf (g + h ) ds →∞ i →∞ i →∞ i i i γ i γ i γ and thus u + v ∈ BVAM (X). For the triangle inequality note that the BVAM upper bounds (gi)and(hi) can be chosen so that for ε>0

DBV u (X) ≥ gi dν − ε, DBV v (X) ≥ hi dν − ε X X for every i. Since (gi +hi)isaBVAM upper bound for u+v, the triangle inequality easily follows from the above property of (gi)and(hi).

Theorem 3.4. The space BVAM (X) with the norm u BV (X) is a Banach space.

Proof. It remains to show the Cauchy property. Let ui, i =1, 2,...,beaCauchy 1 sequence in BVAM (X). Since ui → u in L (X), it suﬃces to show that a subsequence, denoted again by (ui), satisﬁes DBV (ui − u) (X) → 0asi →∞and u ∈ BVAM (X). By the Fuglede lemma (see [5, Lemma 2.1]), we may assume that

(3.4) |ui − u| ds → 0 γ as i →∞for M1–a.e. curve γ in X and

−i (3.5) DBV (ui+1 − ui) (X) < 2

i for each i. From (3.5) it follows that for every i there is a BVAM upper bound (gj ) for ui+1 − ui such that − ≤ i −i (3.6) DBV (ui+1 ui) (X) gj dν < 2 X for all j =1, 2,.... For k =1, 2,... consider the sequence ∞ k l g˜j = gj ,j=1, 2,... . l=k 88 O. MARTIO

k ≥ i ≥ k − ≥ Sinceg ˜j gj for each i k,(˜gj )isaBVAM upper bound for ui+1 ui, i k. k − ≥ Nowg ˜j is a BVAM upper bound for ui+1 uk, i k as well, because i i − ≤ − ≤ l V (ui+1 uk,γ) V (ul+1 ul,γ) lim inf gj ds j→∞ l=k l=k γ i ≤ l ≤ k lim inf gj ds lim inf g˜j ds, j→∞ j→∞ l=k γ γ and this holds for AM–a.e. curve γ by the subadditivity of the AM–modulus. k − Next we show that (˜gj )isaBVAM upper bound for u uk. For this ﬁx a curve γ such that (3.4) holds and that V (u − u ,γ) ≤ lim inf g˜k ds i+1 k →∞ j j γ for every i ≥ k.NotethatAM–a.e. curve γ satisﬁes these conditions. By Lemma 2.1 V (u − uk,γ) ≤ lim inf V (ui+1 − uk,γ) i→∞ ≤ lim inf lim inf g˜k ds = lim inf g˜k ds →∞ →∞ j →∞ j i j γ j γ k and hence (˜g )isaBVAM upper bound for u − uk. From (3.6) it follows that j D (u − u ) (X) ≤ lim inf g˜k dν BV k →∞ j j X ∞ ∞ l ≤ 1 2 (3.7) = lim inf gj dν lim inf = . j→∞ j→∞ 2l 2k l=k X l=k

Thus u − uk ∈ BVAM (X)andsou ∈ BVAM (X). Inequality (3.7) also shows that DBV (u − uk) (X) → 0ask →∞. This completes the proof. In [14] Miranda considered functions u ∈ L1(X) which can be approximated in L1(X) by locally Lipschitz functions whose upper gradients belong to L1(X). Set (3.8) D u (X) = inf{lim inf g dν : u → u in L1(X),u loc. Lipschitz} < ∞ L →∞ i i i i X where gi is an upper gradient of ui and the ﬁrst inﬁmum is taken over all sequences 1 of locally Lipschitz functions (ui)inX such that ui → u in L (X) and all upper gradients gi of ui. The space of all functions such that

u L (X)= u L1(X) + DLu (X) < ∞ is a Banach space and the space is a generalization of the ordinary space of functions of bounded variation in Rn; see [7]. The functions in the above space are also of bounded variation on AM–a.e. curve in X (see [13, Theorem 5.4]) in the sense that V (u, γ) < ∞ on AM–a.e. curve in X. More precisely, if (ui) is a sequence of locally Lipschitz functions with upper 1 gradients gi in X and ui → u in L (X), then there is a subsequence (ui ) such that j | − | → uij u ds 0 γ THE SPACE OF FUNCTIONS ON CURVES IN METRIC MEASURE SPACES 89 on M1–a.e. curve γ in X and (see Lemma 2.3) that V (u, γ) ≤ lim inf V (u ,γ) ≤ lim inf g ds →∞ ij →∞ ij j j γ where gij is an upper gradient of uij . Thus the sequence (gij )isaBVAM upper bound for u. Hence

(3.9) DBV u (X) ≤ DLu (X) holds for every function u ∈ L1(X). Remark 3.5. It would be interesting to know if the inequality

DBV u (Ω) < DLu (Ω) couldholdinsomeopensetΩ⊂ X in the case where functions in L1(X)canbe approximated by locally Lipschitz functions.

4. The measure associated with DBV u

We show that, under some additional assumptions on X,foreachu ∈ BVAM (X) there is a Borel measure μ = μu deﬁned initially on open subsets Ω of X as μ(Ω) = DBV u (Ω) and then extended to arbitrary sets A ⊂ X as μ(A) = inf{μ(Ω) : A ⊂ Ω, Ωopen}. This procedure (see [9]) requires that the space (X, d) is separable and locally compact. These assumptions imply that the metric space X is σ–compact and hence open sets Ω in X can be exhausted by compact sets and, in particular, by open sets with compact closure in Ω. If the measure ν is doubling and the space X is complete, then X is proper, i.e. closed and bounded sets are compact, so that these assumptions imply separability and local compactness. For the discussion on these conditions see [5]. Miranda [14] also used this construction for the corresponding measure based on DLu (Ω). In this section we assume that X is separable and locally compact. The measure ν in X is supposed to satisfy (2.1).

Theorem 4.1. If u ∈ BVAM (X), then the set function μ(Ω) = DBV u (Ω), Ω open in X, deﬁnes a Borel measure in X.

Proof. By [9, Theorem 5.1] it suﬃces to show that for open sets Ω, Ω1 and Ω2 in X it holds that

(4.1) μ(Ω1) ≤ μ(Ω2), if Ω1 ⊂ Ω2,

(4.2) μ(Ω1 ∪ Ω2)=μ(Ω1)+μ(Ω2), if Ω1 ∩ Ω2 = ∅,

(4.3) μ(Ω1 ∪ Ω2) ≤ μ(Ω1)+μ(Ω2),

(4.4) μ(Ω) = sup{μ(Ω1): Ω1 ⊂⊂ Ω}. Here A ⊂⊂ B means that the closure of A is a compact subset of B. Now (4.1) is obvious and (4.2) follows from the fact that each curve in Ω1 ∪ Ω2 lies either in Ω1 or in Ω2.Itremainstoverify(4.3)and(4.4). 90 O. MARTIO

1 2 To prove (4.3) let ε>0andchooseBVAM upper bounds (gi )and(gi )foru in Ω and Ω , respectively, so that 1 2 ≥ j − (4.5) DBV u (Ωj) gi dν ε Ωj 1 \ 2 \ for all i and j =1, 2. Deﬁne gi = gi in Ω1 Ω2, gi = gi in Ω2 Ω1 and gi = 1 2 ∩ max(gi ,gi )inΩ1 Ω2. Then by Lemma 2.4, (gi)isaBVAM upper bound for u in Ω ∪ Ω and we obtain from (4.5) 1 2

DBV u (Ω1 ∪ Ω2) ≤ lim inf gi dν i→∞ Ω1∪Ω2 ≤ 1 2 1 2 lim inf gi dν + gi dν + (gi + gi ) dν i→∞ Ω1\Ω2 Ω2\Ω1 Ω1∩Ω2

≤ DBV u (Ω1)+ DBV u (Ω2)+2ε. Letting ε → 0 we complete the proof for (4.3). To prove (4.4) write

α =sup{μ(Ω1): Ω1 ⊂⊂ Ω}.

We may assume that α<∞.Letε>0 and pick an open set U0 ⊂⊂ Ω such that

(4.6) DBV u (U0) >α− ε.

Next pick open subsets Ui and Vi, i =1, 2,..., such that

U0 ⊂⊂ V1 ⊂⊂ U1 ⊂⊂ V2 ⊂⊂ U2 ⊂⊂ · · · ⊂⊂ Ω ∞ and i=1 Ui = Ω. Note that by (4.6) and (4.1) we have

(4.7) DBV u (U1) >α− ε.

The open sets Ui \ U i−1, i ≥ 2, are disjoint and lie in Ω \ U 1. Hence by (4.2) for each j ≥ 2, j j DBV u ( (Ui \ U i−1)) = DBV u (Ui \ U i−1) i=2 i=2 and we obtain j (4.8) DBV u (U1)+ DBV u ( (Ui \ U i−1))) ≤ α i=2 and thus (4.7) yields j (4.9) DBV u (Ui \ U i−1) <ε i=2 for every j ≥ 2. Similarly we obtain j (4.10) DBV u (Vi+1 \ V i) <ε i=1 for every j ≥ 1. Choose a BV upper bound (g1)foru in U such that AM k 1 1 (4.11) gk dν < α + ε U1 THE SPACE OF FUNCTIONS ON CURVES IN METRIC MEASURE SPACES 91

i and for every i =2, 3,..., a BV upper bound (g )foru in U \ U − such that AM k i i 1 i ε \ − (4.12) gk dν < DBV u (Ui U i 1)+ i Ui\U i−1 2 i for all k =1, 2,... . Similarly for i =1, 2,... we choose BVAM upper bounds (hk) for u in V \ V such that i+1 i i \ ε (4.13) hk dν < DBV u (Vi+1 V i)+ i Vi+1\V i 2 for each k. Since (4.9) and (4.10) hold for each j, (4.12) and (4.13) yield ∞ ∞ i ≤ \ ≤ (4.14) gk dν DBV u (Ui U i−1)+ε 2ε \ i=2 Ui U i−1 i=2 and ∞ ∞ i ≤ \ ≤ (4.15) hk dν DBV u (Vi+1 V i)+ε 2ε. \ i=1 Vi+1 V i i=1 i i Extend each of the functions gk and hk as zero outside their domains of deﬁnition and let ∞ i i g˜k = (gk + hk),k=1, 2,... . i=1 Now (˜gk)isa BVAM upper bound for u inΩ.Toseethisnotethatforaﬁxedk ∞ i i ∈ the sum i=1(gk(x)+hk(x)) contains at most two non-zero terms for each x Ω and that ∞ ∞ Ω=U1 ∪ (Ui \ U i−1) ∪ (Vi+1 \ V i). i=2 i=1 Fix a curve γ in Ω. Since γ([0,l(γ)]) is a compact subset of Ω, there is j such that j j γ([0,l(γ)]) ⊂ U1 ∪ (Ui \ U i−1) ∪ (Vi+1 \ V i)=Vj+1. i=2 i=1 On the other hand, ≥ 1 i i j g˜k(x) gk(x)+ max gk(x)+ max hk(x)=w (x) 2≤i≤j 1≤i≤j k ∈ j for each x Vj+1 and every k and, since by Lemma 2.4, (wk)isaBVAM upper bound for u in Vj+1,(˜gk)isaBVAM upper bound for u in Ω as required. Consequently,

DBV u (Ω) ≤ lim inf g˜k dν →∞ k Ω ∞ ∞ ≤ lim inf g1 dν + gi dν + hi dν →∞ k k k k U U \U − V \V − 1 i=2 i i 1 i=1 i i 1 ∞ ∞ ≤ 1 \ \ lim inf gk dν + DBV u (Vi V i−1)+ DBV u (Vi V i−1) k→∞ U1 i=2 i=1 ≤ α + ε +2ε +2ε = α +5ε where we have used (4.11), (4.14) and (4.15). Since ε>0 is arbitrary, DBV u (Ω) ≤ α and the proof for (4.4) follows and completes the proof for the theorem. 92 O. MARTIO

n 5. The measure DBV u in R n We show that for X = R the space BVAM (X) is the ordinary space of BV functions. The simplest domain Ω in Rn from the modulus and the Fubini theorem point of view is the Cartesian product Ω = (a1,b1) × (a2,b2) ×···×(an,bn). We let Ω =(a1,b1)×(a2,b2)×···×(an−1,bn−1)and(a, b)=(an,bn)sothatΩ=Ω×(a, b). There is nothing particular in the choice of the last coordinate. The classical space BV (Ω) of BV functions in Ω consists of all functions u ∈ L1(Ω) such that { ∇· ∈ 1 n | |≤ } ∞ (5.1) Du (Ω) = sup u ϕdmn : ϕ C0 (Ω, R ), ϕ 1 < ; Ω see [7, Section 5.1]. Note that Lipschitz functions with compact support can be 1 n considered as well as functions ϕ in C0 (Ω, R ). Now Du (U)andU an open subset of Ω, defines a Borel measure in Ω. In the following we let X =Ωandν = mn be the Lebesgue measure and consider the behavior of V (u, γ) on line segments and the measure μ = μu = DBV u , u ∈ L1(Ω), defined in Section 4. For y ∈ Ω let γy(t)=y +(a + t)en, t ∈ [0,b− a], where en =(0, 0,...,1). Then γy is a straight line segment parallel to the xn–axis. Note that γy is a curve 1 in Ω × [a, b] but not in Ω. However, for u ∈ L (Ω), V (u, γy) is well defined; see Section 2. The following lemma shows that the behavior of u is similar on γy as on curves in Ω.

1 Lemma 5.1. Let u ∈ L (Ω), U an open subset of Ω and (gi) a BVAM upper bound for u in U × (a, b) with lim inf g dm < ∞. →∞ i n i U×(a,b)

Then for mn−1–a.e. y ∈ U we have

V (u, γy) ≤ lim inf gi ds < ∞. i→∞ γy ∈ − j ∈ − − Proof. For y U and j>(b a)/2letγy(t)=y+(a+1/j +t)en, t [0,b a 2/j]. j { j } × Then each curve in the family Γ = γy lies in U (a, b) and hence for AM–a.e. j ∈ j γy Γ we have j ≤ ∞ (5.2) V (u, γy) lim inf gi ds < . i→∞ j γy

j Now (5.2) holds for mn−1–a.e. y ∈ U because AM(Γ )=mn−1(U); see [13, Example 3.9]. The Fatou lemma yields

(lim inf gi ds) dmn−1(y) ≤ lim inf gi dmn < ∞ i→∞ i→∞ U γy U×(a,b) and hence j ≤ ∞ V (u, γy) lim inf gi ds < i→∞ γy THE SPACE OF FUNCTIONS ON CURVES IN METRIC MEASURE SPACES 93 for m − –a.e. y ∈ U for each j.Fromthisitfollowsthat n 1 j ≤ ∞ V (u, γy) = lim V (u, γy) lim inf gi ds < j→∞ i→∞ γy as required.

Lemma 5.2. Suppose that u ∈ BVAM (Ω). Then the function y → V (u, γy) belongs 1 to L (Ω ) with respect to the m − measure and n 1

(5.3) V (u, γy) dmn−1 ≤ DBV u (Ω). Ω

Proof. We ﬁrst show that the function V (u, γy)=V (u◦ γy, [0,b−a]) is measurable in Ω . To see this note that u◦ γy(t)=u(y, γ(t)) for a.e. (y, t) ∈ Ω × [0,b− a] and thus u◦ γy is a measurable function in Ω × [0,b − a] and, moreover, an 1 L (Ω × [0,b− a])–function. Pick a set C ⊂ [0,b− a] such that m1([0,b− a] \ C)=0 and for every t ∈ C the function y → u◦ γy(t)ismn−1–measurable in Ω .Thisis possible by the Fubini theorem. Thus if the points t1 0andchooseaBV upper bound (g )foru in Ω with AM i lim inf g dm ≤ D u (Ω) + ε. →∞ i n BV i Ω By Lemma 5.1 and the Fatou lemma we have

V (u, γy) dmn−1 ≤ (lim inf gi ds) dmn−1 i→∞ Ω Ω γy ≤ lim inf g dm ≤ D u (Ω) + ε, →∞ i n BV i Ω and letting ε → 0 we obtain (5.3).

The measure DBV u deﬁnes a Borel measureμ ˆ in Ω by the ruleμ ˆ(A)= DBV u (A × (a, b)) for each Borel set A ⊂ Ω . The next theorem clariﬁes the roles of V (u, γy)andˆμ. Lemma 5.3. Let u ∈ BVAM (Ω). Then for mn−1–a.e. y ∈ Ω

(5.5) V (u, γy) ≤ dμˆ(y) n−1 n−1 where dμˆ(y) = limr→0 μˆ(B (y, r))/(mn−1(B (y, r)) is the derivative of μˆ with respect to the mn−1 measure at y ∈ Ω . 94 O. MARTIO

Proof. By Lemma 5.1 for m − –a.e. y ∈ Ω , n 1 (5.6) V (u, γ ) ≤ lim inf g ds y →∞ i i γ n−1 n−1 whenever (gi)isaBVAM upper bound for u in B (z,r) × (a, b), B (z,r) ⊂ Ω and, by Lemma 5.2, the Lebesgue theorem yields n−1 −1 (5.7) V (u, γ ) = lim m − (B (y, r)) V (u, γ ) dm − (x) y → n 1 x n 1 r 0 Bn−1(y,r) for mn−1–a.e. y ∈ Ω . Fix a point y ∈ Ω such that (5.6) and (5.7) hold and the derivative dμˆ(y) exists. Almost every point y ∈ Ω satisﬁes these conditions. Choose radii r1 >r2 > ... n−1 with limj→∞ rj =0,setBj = B (y, rj )) and for each j pick a BVAM upper j × bound (gi )foru in Bj (a, b) such that rn−1 j ≤ × j (5.8) gi dmn DBV u (Bj (a, b)) + j Bj ×(a,b) 2 for every i =1, 2,... . The Fatou lemma and (5.8) yield −1 V (u, γ ) = lim m − (B ) V (u, γ ) dm − (x) y →∞ n 1 j x n 1 j B j −1 j ≤ lim inf mn−1(Bj) (lim inf g ds) dmn−1(x) j→∞ i→∞ i Bj γx −1 j ≤ lim inf mn−1(Bj) lim inf g dmn j→∞ i→∞ i Bj ×(a,b) n−1 −1 rj ≤ lim inf mn−1(Bj ) ( DBV u (Bj × (a, b)) + )=dμˆ(y) j→∞ 2j and (5.5) follows.

The next theorem shows that the spaces BVAM (Ω) and BV (Ω) deﬁned in Section 3 and in (5.1), respectively, are the same. Theorem 5.4. For each u ∈ L1(Ω),

(5.9) DBV u (Ω) ≤ Du (Ω) ≤ n DBV u (Ω). Remark 5.5. It follows from the proof that the ﬁrst inequality in (5.9) holds for all open sets U ⊂ Ω. ∈ ∞ Proof. Let u BV (Ω). By [7, 5.2.2, Theorem 2] there is a sequence of C0 (Ω)– functions u such that u → u in L1(Ω) and i i Du (Ω) = lim |∇u | dm . →∞ i n i Ω Then passing to a subsequence (see Lemma 2.3) we can assume that the sequence (|∇u |)isaBV upper bound for u in Ω. Hence i AM D u (Ω) ≤ lim inf |∇u | dm = Du (Ω) BV →∞ i n i Ω and the ﬁrst inequality in (5.9) follows. THE SPACE OF FUNCTIONS ON CURVES IN METRIC MEASURE SPACES 95

If u ∈ BV (Ω), then there exist measures Dj u , j =1, 2,..., n, deﬁned on ⊂ open sets U Ωas

Dj u (U)=sup{ u∂j ϕdmn} U ∈ ∞ | |≤ where the supremum is taken over all ϕ C0 (U) such that ϕ 1; see [7, 5.10]. By [7, the proof for Theorem 2 (7.10.2)] and Lemma 5.2 we have

Dnu (Ω) = V (u, γy) dmn−1(y) ≤ DBV u (Ω) Ω and a similar inequality holds for each j =1, 2,..., n.Letε>0andchoosea 1 n | |≤ function ϕ =(ϕ1,ϕ2,...,ϕn)inC0 (Ω, R ) such that ϕ 1and

Du (Ω) ≤ u∇·ϕdmn + ε. Ω Now the above estimates yield n n Du (Ω) ≤ u∂j ϕj dmn + ε ≤ Dj u (Ω) + ε ≤ n DBV u (Ω) + ε, j=1 Ω j=1 and letting ε → 0 we obtain the second inequality in (5.9).

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Department of Mathematics and Statistics, FI–00014 University of Helsinki, Finland E-mail address: [email protected]