Functions of bounded variation on complete and connected one-dimensional metric spaces
aa r X i v : . [ m a t h . M G ] S e p FUNCTIONS OF BOUNDED VARIATION ON COMPLETE ANDCONNECTED ONE-DIMENSIONAL METRIC SPACES
PANU LAHTI AND XIAODAN ZHOU
Abstract.
In this paper, we study functions of bounded variation on a complete andconnected metric space with finite one-dimensional Hausdorff measure. The definitionof BV functions on a compact interval based on pointwise variation is extended to thisgeneral setting. We show this definition of BV functions is equivalent to the BV functionsintroduced by Miranda [18]. Furthermore, we study the necessity of conditions on theunderlying space in Federer’s characterization of sets of finite perimeter on metric measurespaces. In particular, our examples show that the doubling and Poincar´e inequalityconditions are essential in showing that a set has finite perimeter if the codimension oneHausdorff measure of the measure-theoretic boundary is finite. Introduction
Functions of bounded variation, also known as BV functions, have been extensivelystudied and widely applied in different areas including the calculus of variations, hyperbolicconservation laws, and minimal surfaces [3, 6, 9]. In the context of metric measure spaces,the notion of functions of bounded variation is introduced by Miranda [18] and it hasattracted significant attention in recent years (e.g. [1, 2, 13, 16, 17]). Motivated by theobservation that various function classes including Sobolev functions and BV functionsdefined on the real line R have simple characterizations, in this work we focus our study onBV functions in one-dimensional metric spaces. Our main result gives a simple alternativedefinition of BV functions in a general one-dimensional space based on pointwise variation.Let Ω denote an open set in the Euclidean space R n . A function u ∈ L (Ω) is said tohave bounded variation in Ω if k Du k (Ω) := sup (cid:26) ˆ Ω u div ϕ dx : ϕ ∈ C c (Ω; R n ) , | ϕ | ≤ (cid:27) < ∞ . By the Riesz representation theorem, the class of functions with bounded variation in Ω,denoted by BV(Ω), is the collection of functions whose weak first partial derivatives areRadon measures. An equivalent characterization of BV functions is given as the L lim-its of sequences of smooth functions with gradients bounded in L . By replacing smoothfunctions with locally Lipschitz functions and the absolute value of the gradient by a local Mathematics Subject Classification.
Key words and phrases.
Function of bounded variation, one-dimensional metric space, Federer’s charac-terization of sets of finite perimeter.
1V FUNCTIONS ON ONE-DIMENSIONAL SPACES 2
Lipschitz constant, Miranda [18] introduced functions of bounded variation on a completedoubling metric measure space (
X, d, µ ) supporting a Poincar´e inequality. Equivalent def-initions of BV functions on complete and separable metric measure spaces are studied byAmbrosio and Di Marino [2]. They relax the locally Lipschitz functions in Miranda’s defi-nition to a more general class of functions, with the local Lipschitz constants replaced byupper gradients. We recall the definition of BV functions on general metric measure spacesusing upper gradients.
Definition 1.1.
Given an open set Ω ⊂ X and a function u on Ω, the total variation of u in Ω is defined by k Du k (Ω) := inf (cid:26) lim inf i →∞ ˆ Ω g u i dµ : u i → u in L (Ω) (cid:27) , where each g u i is an upper gradient of u i in Ω. A function u is said to have boundedvariation on Ω if k Du k (Ω) < ∞ .On the real line R , various function classes usually have simpler characterizations. Forexample, upon choosing a good representative, we can identify a Sobolev function u ∈ W ,p ([ a, b ]) with an absolutely continuous function with p -integrable derivative [7, Theorem1, Page 163]. Functions of bounded variation on R can also be characterized by pointwisevariation. Recall that the pointwise variation of a function u : [ a, b ] → R is defined asPV( u, [ a, b ]) := sup ( n − X k =1 | u ( t k ) − u ( t k +1 ) | , a ≤ t ≤ . . . ≤ t n ≤ b ) . (1.1)If Ω ⊂ R is open, the pointwise variation PV( u, Ω) is defined as P I PV( u, I ), where thesum runs along all the closed intervals in Ω. The essential variation eV( u, Ω) is defined aseV( u, Ω) := inf { PV( v, Ω) : u = v a.e. in Ω } . For u ∈ L (Ω), we have eV( u, Ω) = k Du k (Ω) [3, Theorem 3.27].The above characterizations of function classes can be extended to general one-dimensionalmetric spaces. Let X be a complete and connected metric space with finite one-dimensionalHausdorff measure H ( X ) < ∞ . In [19], the notion of absolutely continuous functions isgeneralized and Newtonian Sobolev functions are characterized by these absolutely contin-uous functions. Functions of bounded variations on curves in metric measure spaces arestudied by Martio [16, 17]. In this work, we investigate the pointwise variation characteri-zations of BV functions on the above one-dimensional space. We first give the definition: Definition 1.2.
Let X be a complete connected metric measure space with H ( X ) < ∞ .For a function v on X , we define the pointwise variation aspV( v, X ) := sup X j | v ◦ γ j ( ℓ j ) − v ◦ γ j (0) | , V FUNCTIONS ON ONE-DIMENSIONAL SPACES 3 where the supremum is taken over all finite collections of pairwise disjoint injective arc-length parametrized curves γ j : [0 , ℓ j ] → X . Then we defineVar( u, X ) := inf { pV( v, X ) , v = u a.e. on X } . A function u : X → R has bounded pointwise variation if Var( u, X ) < ∞ .It can be shown that when X is an interval, we have Var( u, X ) = eV( u, X ). Remark 1.1.
In the above definition, one could replace | v ◦ γ j ( ℓ j ) − v ◦ γ j (0) | with PV( v ◦ γ j , [0 , ℓ j ]) for each simple curve. Lemma 3.1 shows that the two quantities are comparable.We say that a function e u is a good representative of u if u = e u almost everywhere andVar( u, X ) = pV( e u, X ). We show that every function u with Var( u, X ) < ∞ admits a goodrepresentative. Lemma 1.1 (Existence of a good representative) . Suppose that ( X, d, H ) is a completeand connected metric measure space with H ( X ) < ∞ . If Var( u, X ) < ∞ , then there existsa function e u on X with e u = u a.e. and pV( e u, X ) = Var( u, X ) = inf { pV( v, X ) : v = u a.e. on X } . We show that the class of BV functions given by Definition 1.2 is equivalent to the BVfunctions given in Definition 1.1. The main theorem is stated below:
Theorem 1.1 (Main Theorem) . Suppose that ( X, d, H ) is a complete and connected met-ric measure space with H ( X ) < ∞ . Let u be a function on X . Then the following hold: (1) If k Du k ( X ) < ∞ , then Var( u, X ) ≤ k Du k ( X ) . (2) Suppose there exists a constant C such that for all x ∈ X lim inf r → H ( B ( x, r )) r < C (1.2) holds. If Var( u, X ) < ∞ , then k Du k ( X ) < ∞ . Remark 1.2.
In particular, if X is complete, connected and Ahlfors 1-regular with H ( X ) < ∞ , a function u on X satisfies k Du k ( X ) < ∞ if and only if Var( u, X ) < ∞ . Remark 1.3.
The density upper bound (1.2) turns out to be essential in this characteri-zation. Complete and connected metric spaces (
X, d ) with H ( X ) < ∞ can be constructedsuch that a function u satisfies k Du k ( X ) = ∞ while Var( u, X ) < ∞ , see Example 4.1 andExample 4.2.The proof for the first part of the main theorem is standard and is given in Proposition3.1. The second part requires a more delicate argument. Suppose u is a function withVar( u, X ) < ∞ . We first use the existence of good representatives to show that Var( v, X )is lower semicontinuous with respect to convergence in L ( X ). Then we prove the coarea V FUNCTIONS ON ONE-DIMENSIONAL SPACES 4 inequality stated below, first for curve-continuous functions, i.e. functions that are contin-uous along every curve in X . A sequence of curve-continuous functions u i approximating u in L ( X ) can be constructed such that the limit superior of pV( u i , X ) is bounded above by C Var( u, X ), where C is a constant. These facts imply the following result; χ E denotesthe characteristic function of E ⊂ X . Lemma 1.2 (Co-area Inequality) . Let ( X, d, H ) be a complete and connected metric mea-sure space with H ( X ) < ∞ . Suppose there exists a constant C such that for all x ∈ X lim inf r → H ( B ( x, r )) r < C holds. Suppose Var( u, X ) < ∞ . Then C Var( u, X ) ≥ ˆ ∗ R Var( χ { u>t } , X ) dt. Using also the BV coarea formula [18, Proposition 4.2] (see detailed statement (2.4) inSection 2), it now suffices to consider u = χ E for Var( χ E , X ) < ∞ . Hence the proof iscompleted by showing that k Dχ E k ( X ) is bounded above by C Var( χ E , X ).An interesting and important aspect of the theory of BV functions lies in the analysis ofsets of finite perimeter, that is, sets whose characteristic functions are BV functions. For aset E ⊂ R n , Federer’s characterization of sets of finite perimeter [8] states that E has finiteperimeter if and only if the codimension one Hausdorff measure of its measure-theoreticboundary satisfies H ( ∂ ∗ E ) < ∞ , see Section 4 for detailed definitions. Let ( X, d, µ ) bea complete and doubling metric measure space that supports a 1-Poincar´e inequality andlet E ⊂ X be a measurable set. Ambrosio [1, Theorem 5.3] shows that if E has finiteperimeter then H ( ∂ ∗ E ) < ∞ . The converse implication of Federer’s characterization inthe general metric space setting is proved by the first author in [15, Theorem 1.1].It has not been known so far whether the doubling and Poincar´e inequality conditions onthe underlying space are necessary when showing that the condition H ( ∂ ∗ E ) < ∞ impliesthat E is of finite perimeter. By constructing simple explicit examples of one-dimensionalspaces, we show that these two conditions are really essential.This paper is organized in the following way: preliminaries are covered in Section 2 andthe proof of the main theorem is presented in Section 3. In Section 4, we construct twoexamples to show the necessity of the doubling condition and the Poincar´e inequality inFederer’s characterization. 2. Definitions and notation
Assume throughout the paper that (
X, d, H ) is a complete and connected metric spacewith H ( X ) < ∞ . If a property holds outside a set of H -measure zero, we say that itholds almost everywhere, abbreviated a.e. The symbol C will denote a constant that onlydepends on the space X . We say that a measure µ is doubling if there exists a constant V FUNCTIONS ON ONE-DIMENSIONAL SPACES 5 C such that µ ( B ( x, r )) ≤ Cµ ( B ( x, r )) for all open balls B ( x, r ). The space X is Ahlfors s -regular if there is a constant C such that C − r s ≤ µ ( B ( x, r )) ≤ Cr s , whenever x ∈ X and 0 < r < diam( X ). If X is Ahlfors s -regular with respect to µ , we canreplace µ by the s -dimensional Hausdorff measure H s without losing essential information[12, Exercise 8.11].A continuous mapping γ : [ a, b ] → X is said to be a rectifiable curve if it has finitelength. A rectifiable curve always admits an arc-length parametrization (see e.g. [10,Theorem 3.2]). If γ : [ a, b ] → X is a rectifiable curve and g : γ ([ a, b ]) → [0 , ∞ ] is a Borelfunction, we define ˆ γ g ds := ˆ ℓ g ( e γ ( s )) ds, where e γ : [0 , ℓ ] → X is the arc-length parametrization of γ . From now on we will assumeall curves to be rectifiable and arc-length parametrized unless otherwise specified. Definition 2.1 (Upper gradient) . Let u : X → R . We say that a Borel function g : X → [0 , ∞ ] is an upper gradient of u if | u ( γ ( ℓ γ )) − u ( γ (0)) | ≤ ˆ γ g ds (2.1)for every curve γ . We use the conventions ∞ − ∞ = ∞ and ( −∞ ) − ( −∞ ) = −∞ . If g : X → [0 , ∞ ] is a µ -measurable function and (2.1) holds for 1-almost every curve, we saythat g is a 1-weak upper gradient of u . A property is said to hold for 1-almost every curveif there exists ρ ∈ L ( X ) such that ´ γ ρ ds = ∞ for every curve γ for which the propertyfails.For 1 ≤ p < ∞ , the Newtonian Sobolev class N ,p ( X ) consists of those L p -integrablefunctions on X for which there exists a p -integrable upper gradient.The notation u B stands for an integral average, that is, u B := B u dµ := 1 µ ( B ) ˆ B u dµ. A metric measure space supporting a Poincar´e inequality is defined in the following way.
Definition 2.2 (Space supporting Poincar´e inequality) . Let 1 ≤ p < ∞ . A metric measurespace ( X, d, µ ) is said to support a p-Poincar´e inequality if there exists constants
C > λ ≥ u : X → R and g : X → [0 , ∞ ],where u is measurable and g is an upper gradient of u : B ( x,r ) | u − u B ( x,r ) | dµ ≤ Cr B ( x,λr ) g p dµ ! p for every ball B ( x, r ). V FUNCTIONS ON ONE-DIMENSIONAL SPACES 6
A metric space X is quasiconvex if every two points can be joined by a curve withlength comparable to the distance between these two points. If X is complete, doublingand supports a p -Poincar´e inequality for 1 ≤ p < ∞ , then X is quasiconvex [11, Proposition4.4].We recall the following generalization of the Euclidean area formula to the case of Lips-chitz maps f from the Euclidean space R n into a metric space X . The proof can be foundin [14, Corollary 8]. Theorem 2.1 (Area formula) . Let f : R n → X be Lipschitz. Then ˆ R n g ( x ) J n ( mdf x ) dx = ˆ X X x ∈ f − ( y ) g ( x ) d H n ( y ) for any Borel function g : R n → [0 , ∞ ] , and ˆ A g ( f ( x )) J n ( mdf x ) dx = ˆ X g ( y ) H ( A ∩ f − ( y )) d H n ( y ) for A ⊂ R n measurable and any Borel function g : X → [0 , ∞ ] . We apply the above theorem to an arc-length parametrized curve. Let f = γ and γ : [0 , ℓ ] → X . In this case, J ( mdf x ) equals the metric derivative defined as | ˙ γ | ( t ) := lim h → d ( γ ( t + h ) , γ ( t )) | h | , and | ˙ γ | ( t ) = 1 for almost every t ∈ [0 , ℓ ]. Let Γ = γ ([0 , ℓ ]) and let g : X → [0 , ∞ ] be a Borelfunction. It follows from Theorem 2.1 that ˆ ℓ g ( γ ( s )) ds = ˆ Γ g ( y ) H ([0 , ℓ ] ∩ γ − ( y )) d H ( y ) . (2.2)A compact and connected 1-dimensional metric space admits a nice parametrization.The proofs of the following two classical results can be found in [4, Theorem 4.4.7, Theorem4.4.8]. Theorem 2.2 (First Rectifiability Theorem) . If E is complete and C ⊂ E is a closedconnected set such that H ( C ) < ∞ , then C is compact and connected by simple curves. Theorem 2.3 (Second Rectifiability Theorem) . If E is complete, C ⊂ E is closed andconnected, and H ( C ) < ∞ , then there exist countably many arc-length parametrized simplecurves γ i : [0 , ℓ i ] → C such that H (cid:16) C \ ∞ [ i =1 γ i ([0 , ℓ i ]) (cid:17) = 0 . Given u ∈ Lip loc ( X ), we define the local Lipschitz constant byLip u ( x ) := lim sup y → x | u ( y ) − u ( x ) | d ( y, x ) . (2.3) V FUNCTIONS ON ONE-DIMENSIONAL SPACES 7
Given an open set Ω ⊂ X and a function u ∈ L (Ω), we define the total variation of u in Ω by k Du k (Ω) := inf (cid:26) lim inf i →∞ ˆ Ω g u i dµ : u i ∈ N , (Ω) , u i → u in L (Ω) (cid:27) , where each g i is a (1-weak) upper gradient of u i in Ω. We say that a function u ∈ L (Ω)is of bounded variation, and denote u ∈ BV(Ω), if k Du k (Ω) < ∞ . A µ -measurable set E ⊂ X is said to be of finite perimeter if k Dχ E k ( X ) < ∞ , where χ E is the characteristicfunction of E .The following coarea formula is given in [18, Proposition 4.2]: if Ω ⊂ X is an open setand u ∈ L (Ω), then k Du k (Ω) = ˆ ∗ R k Dχ { u>t } k (Ω) dt, (2.4)where we abbreviate { u > t } := { x ∈ Ω : u ( x ) > t } . We use an upper integral sincemeasurability is not clear, but if either side is finite, then both sides are finite and we alsohave measurability. 3. Proofs of the main results
Standing assumptions:
We will assume throughout this section that (
X, d, H ) isa complete and connected metric measure space with 0 < H ( X ) < ∞ . By the FirstRectifiability Theorem 2.2, it follows that X is compact.3.1. Finite total variation implies finite pointwise variation.
We prove part (1) ofTheorem 1.1 first.
Proposition 3.1.
Let u be a function on X such that k Du k ( X ) < ∞ . Then Var( u, X ) ≤k Du k ( X ) .Proof. From the definition of the total variation we find a sequence ( u i ) such that u i → u in L ( X ) and lim i →∞ ˆ X g i d H = k Du k ( X ) , (3.1)where each g i is an upper gradient of u i . Passing to a subsequence (not relabeled), we alsohave u i → u a.e. By the First Rectifiability Theorem 2.2, for every pair of points x, y ∈ X we find a simple curve γ : [0 , ℓ ] → X with γ (0) = x and γ ( ℓ ) = y , and then by (2.2), | u i ( y ) − u i ( x ) | ≤ ˆ γ g i ds ≤ ˆ X g i d H → k Du k ( X ) as i → ∞ . Thus the functions u i are uniformly bounded. Note that the sequence of Radon measures g i d H has uniformly bounded mass, and so we know that passing to a subsequence (notrelabeled) we have g i d H ∗ ⇀ dν for some Radon measure ν on X [3, Theorem 1.59]. This V FUNCTIONS ON ONE-DIMENSIONAL SPACES 8 reference also gives the lower semicontinuity ν ( X ) ≤ lim i →∞ ˆ X g u i d H = k Du k ( X ) . (3.2)Moreover, for any compact set K ⊂ X we have ν ( K ) ≥ lim sup i →∞ ˆ K g i d H ; (3.3)see [3, Proposition 1.62] (and then in fact equality holds in (3.2)). Define v ( x ) := lim sup i →∞ u i ( x )for every x ∈ X , so that v = u H -a.e., and v is bounded since the functions u i are uniformlybounded. Now for every simple curve γ : [0 , ℓ ] → X we have | v ◦ γ ( ℓ ) − v ◦ γ (0) | ≤ lim sup i →∞ | u i ◦ γ ( ℓ ) − u i ◦ γ (0) |≤ lim sup i →∞ ˆ γ g i ds = lim sup i →∞ ˆ γ ([0 ,ℓ ]) g i d H by (2.2) ≤ ν ( γ ([0 , ℓ ])) by (3.3) . It follows that for any finite collection of pairwise disjoint simple curves γ j : [0 , ℓ j ] → X , X j | v ◦ γ j ( ℓ j ) − v ◦ γ j (0) | ≤ X j ν ( γ j ([0 , ℓ j ])) ≤ ν ( X ) ≤ k Du k ( X ) by (3.2) . It follows that pV( v, X ) ≤ k Du k ( X ) and so Var( u, X ) ≤ k Du k ( X ). (cid:3) Finite pointwise variation implies finite total variation.
The proof of part (2)of Theorem 1.1 is more involved. We divide the argument into several parts.3.2.1.
Existence of a good representative.
We first show that every u with Var( u, X ) < ∞ admits a good representative e u . As a result, Var( u, X ) turns out to be lower semicontinuouswith respect to convergence in L ( X ).Note that we can define an alternative version of the pointwise variation of a function v on X by PV( v, X ) := sup X j PV( v ◦ γ j ) , where the supremum is taken over finite collections of pairwise disjoint simple curves γ j : [0 , ℓ j ] → X , and we denote PV( v ◦ γ j ) := PV( v ◦ γ j , [0 , ℓ j ]); recall (1.1). Then ob-viously pV( v, X ) ≤ PV( v, X ). Conversely, we have the following.
Lemma 3.1.
For any function v on X , we have PV( v, X ) ≤ v, X ) .Proof. Consider a simple curve γ . Take a partition 0 = t ≤ t ≤ . . . ≤ t n = ℓ γ . Suppose n is odd (the case of even n is similar). Then the subcurves γ | [ tk,tk +1] , for k = 0 , , . . . , n − V FUNCTIONS ON ONE-DIMENSIONAL SPACES 9 are disjoint, and so are the subcurves γ | [ tk,tk +1] for k = 1 , , . . . , n −
2. Let γ k be γ | [ tk,tk +1] reparametrized by arc-length. Then n − X k =0 | v ( γ ( t k )) − v ( γ ( t k +1 )) | = X k =0 , ,...,n − | v ( γ k (0)) − v ( γ k ( ℓ γ k )) | + X k =1 , ,...,n − | v ( γ k (0)) − v ( γ k ( ℓ γ k )) | . Taking supremum over all partitions, we get PV( v ◦ γ, [0 , ℓ γ ]) ≤ v, X ). If we considercollections of pairwise disjoint simple curves γ j , and if we do the above for each γ j , weobtain that PV( v, X ) ≤ v, X ). (cid:3) Next we show that we can find a good representative e u of any function u , with pV( e u, X ) =Var( u, X ). In proving this we will take inspiration from Martio [16]. Given a function v on X and a set D ⊂ X , we definepV D ( v, X ) := sup X j | v ◦ γ j ( ℓ j ) − v ◦ γ j (0) | , where the supremum is taken over finite collections of pairwise disjoint simple curves γ j : [0 , ℓ j ] → X with endpoints γ j (0) , γ j ( ℓ j ) ∈ D . Proposition 3.2.
Let D ⊂ X be an arbitrary set with H ( X \ D ) = 0 . Suppose pV D ( v, X ) < ∞ . Then there exists a function v e on X such that v e = v on D and pV( v e , X ) =pV D ( v, X ) .Proof. If x ∈ D , define v e ( x ) = v ( x ). Fix a point z ∈ D . For any point x ∈ X \ D , bythe First Rectifiability Theorem (Theorem 2.2), there exists a simple curve γ x : [0 , ℓ x ] → X with γ x (0) = x and γ x ( ℓ x ) = z . We define v e ( x ) := lim t → + , γ x ( t ) ∈ D v ◦ γ x ( t ) . The limit exists since the quantitysup ( n − X k =1 | v ◦ γ x ( t k ) − v ◦ γ x ( t k +1 ) | , ≤ t ≤ . . . ≤ t n ≤ ℓ x , γ x ( t k ) ∈ D ) is finite, which follows from the condition pV D ( v, X ) < ∞ just as in Lemma 3.1. Thenwe show that v e : X → R , with v e = v on D , satisfies pV( v e , X ) = pV D ( v, X ). It is clearthat pV( v e , X ) ≥ pV D ( v, X ). Conversely, let { γ j } nj =1 be an arbitrary collection of pairwisedisjoint curves. If all the endpoints γ j (0) , γ j ( ℓ j ) ∈ D , then n X j =1 | v ◦ γ j ( ℓ j ) − v ◦ γ j (0) | = n X j =1 | v e ◦ γ j ( ℓ j ) − v e ◦ γ j (0) | . V FUNCTIONS ON ONE-DIMENSIONAL SPACES 10
If there exists a point p j = γ j ( ℓ j ) ∈ X \ D (or γ j (0) ∈ X \ D , or both), then we denote thecurve connecting z and p j in the definition of the function value of v e at p j by γ p j : [0 , ℓ p j ] → X . Let ǫ > δ > γ j intersects with γ p j only at p j inside B ( p j , δ ), thenwe define a simple curve e γ j : [0 , e ℓ j ] → X by e γ j ( t ) := γ j ( t ) if 0 ≤ t ≤ ℓ j γ p j ( t − ℓ j ) if ℓ j ≤ t ≤ e ℓ j where e ℓ j ≤ ℓ j + δ . By choosing e ℓ j sufficiently close to ℓ j , we have that | v ◦ e γ j ( e ℓ j ) − v e ◦ γ j ( ℓ j ) | < ǫ n . Likewise, if p j = γ j (0) ∈ X \ D , we can also extend γ j slightly to e γ j by attachinga small piece of γ p j at the endpoint such that | v ◦ e γ j (0) − v e ◦ γ j (0) | < ǫ n . (2) If for every δ > q ∈ B ( p j , δ ) with q = p j such that q = γ j ( e t ) = γ p j ( t )for some e t, t , then we define e γ j : [0 , e ℓ j ] → X as the restriction of γ j to [0 , e t ], so that | v ◦ e γ j ( e ℓ j ) − v e ◦ γ j ( ℓ j ) | = | v ◦ γ j ( e t ) − v e ◦ γ j ( ℓ j ) | = | v ◦ γ p j ( t ) − v e ( p j ) |≤ ǫ n , if we choose t sufficiently close to 0. A similar modification works for the case when p j = γ j (0) . Then we get a new collection of curves { e γ j } nj =1 defined as above if at least one of theendpoints of γ j belong to X \ D . Furthermore, since the curves γ j are pairwise disjoint,we can choose δ sufficiently small such that the curves e γ j are pairwise disjoint. Hence, weget that n X j =1 | v e ◦ γ j ( ℓ j ) − v e ◦ γ j (0) | ≤ n X j =1 | v ◦ e γ j ( e ℓ j ) − v ◦ e γ j (0) | + ǫ. This implies that pV( v e , X ) ≤ pV D ( v, X ), and pV( v e , X ) = pV D ( v, X ) follows. (cid:3) Proposition 3.3.
Suppose
Var( u, X ) < ∞ . Then there exists a function e u on X with e u = u a.e. and pV( e u, X ) = Var( u, X ) = inf { pV( v, X ) : v = u a.e. on X } . Proof.
Take a function v = u a.e. with pV( v, X ) < ∞ . Let u i : X → R be a sequence suchthat u i = v on D i with H ( X \ D i ) = 0 and pV( v i , X ) → Var( u, X ). Let D := T i D i .Then u i = v on D and H ( X \ D ) = 0. By Proposition 3.2 there exists e u : X → R such V FUNCTIONS ON ONE-DIMENSIONAL SPACES 11 that e u = v on D andpV( e u, X ) = pV D ( v, X ) = pV D ( u i , X ) ≤ pV( u i , X ) → Var( u, X ) as i → ∞ . (cid:3) We have the following lower semicontinuity results.
Proposition 3.4.
Suppose D ⊂ X and v i ( x ) → v ( x ) for all x ∈ D . Then pV D ( v, X ) ≤ lim inf i →∞ pV D ( v i , X ) . Next suppose u i → u in L ( X ) . Then Var( u, X ) ≤ lim inf i →∞ Var( u i , X ) . Proof.
The first claim is easy to check. To prove the second, we can assume that the right-hand side is finite and in fact that Var( u i , X ) < ∞ for each i ∈ N , and then we can choosegood representatives e u i . Passing to a subsequence (not relabeled) we have e u i ( x ) → u ( x )for every x ∈ D with H ( X \ D ) = 0. By the first claim,pV D ( u, X ) ≤ lim inf i →∞ pV D ( e u i , X ) ≤ lim inf i →∞ pV( e u i , X )= lim inf i →∞ Var( u i , X ) < ∞ . (3.4)By Proposition 3.2, there exists an extension u e for u restricted to D satisfying u e = u on D and pV( u e , X ) = pV D ( u, X ). In particular, u e = u a.e. on X . We getVar( u, X ) = inf { pV( v, X ) : v = u a.e. on X }≤ pV( u e , X )= pV D ( u, X )= lim inf i →∞ Var( u i , X )by (3.4). (cid:3) Approximation by curve-continuous functions.
We say that a function v on X iscurve-continuous if v ◦ γ is continuous for every curve γ in X . In this part, we exploit the niceproperties of curve-continuous functions to show that every function with Var( u, X ) < ∞ is H -measurable and it can be approximated in L ( X ) by a sequence of curve-continuousfunctions u i such that lim sup i →∞ pV( u i , X ) ≤ C Var( u, X )for some constant C depending only on C in the density upper bound condition (1.2).We first show that every curve-continuous function is H measurable. Lemma 3.2.
Let v be a curve-continuous function on X . Then v is H -measurable. V FUNCTIONS ON ONE-DIMENSIONAL SPACES 12
Proof.
Let t ∈ R . It suffices to show that { v ≥ t } is H -measurable. By curve-continuity, foreach curve γ : [0 , ℓ ] → X the set γ ([0 , ℓ ]) ∩ { v ≥ t } is compact. By the Second RectifiabilityTheorem 2.3, there exist curves γ j : [0 , ℓ j ] → X , j ∈ N , such that H X \ ∞ [ j =1 γ j ([0 , ℓ j ]) = 0 . The set S ∞ j =1 ( γ j ([0 , ℓ j ]) ∩ { v ≥ t } ) is a Borel set and differs from { v ≥ t } only by a set of H -measure zero. (cid:3) For a function v on X and t ∈ R , r >
0, we define the truncations v t := min { t, v } and v t,t + r := max { t, min { t + r, v }} . Lemma 3.3.
Let v be a curve-continuous function on X with pV( v, X ) < ∞ and let t ∈ R , r > . Then pV( v t , X ) + pV( v t,t + r , X ) ≤ pV( v t + r , X ) . Proof.
Consider a curve γ used in estimating pV( v t , X ) < ∞ . Note that v t ≡ t in { v ≥ t } .Thus, by also reversing direction if necessary, we can assume that γ (0) ∈ { v < t } . Supposealso γ ( ℓ γ ) ∈ { v < t } , but γ intersects { v ≥ t } . Let s , s be the smallest and largestnumber, respectively, for which γ ( s ) , γ ( s ) ∈ { v ≥ t } ; these exist by the curve-continuity.If ε >
0, by curve-continuity we find e s < s , e s > s such that v t ( γ ( e s )) > t − ε and v t ( γ ( e s )) > t − ε . Then for the subcurves γ := γ | [0 , e s ] and γ := γ | [ e s ,ℓ γ ] (reparametrizedby arc-length) we have | v t ( γ (0)) − v t ( γ ( ℓ γ )) | ≥ | v t ( γ (0)) − t | − ε and | v t ( γ (0)) − v t ( γ ( ℓ γ )) | ≥ | v t ( γ ( ℓ γ )) − t | − ε. Thus | v t ( γ (0)) − v t ( γ ( ℓ γ )) | + | v t ( γ (0)) − v t ( γ ( ℓ γ )) | ≥ | v t ( γ (0)) − v t ( γ ( ℓ γ )) | − ε. Since ε > v, X ), we can replacethe curve γ by two curves that are contained in { v < t } . Similarly, if γ (0) ∈ { v < t } and γ ( ℓ γ ) ∈ { v ≥ t } , we can replace such γ by one subcurve that is in { v < t } .Now fix ε > γ j contained inside { v < t } such that N X j =1 | v t ◦ γ j ( ℓ j ) − v t ◦ γ j (0) | + ε > pV( v t , X ) . V FUNCTIONS ON ONE-DIMENSIONAL SPACES 13
Analogously, we find a collection of pairwise disjoint simple curves γ j contained inside { v > t } such that N X j = N +1 | v t,t + r ◦ γ j ( ℓ j ) − v t,t + r ◦ γ j (0) | + ε > pV( v t,t + r , X ) . Now the curves γ j , j = 1 , . . . , N , are pairwise disjoint, and thuspV( v t , X ) + pV( v t,t + r , X ) ≤ N X j =1 | v t ◦ γ j ( ℓ j ) − v t ◦ γ j (0) | + N X j = N +1 | v t,t + r ◦ γ j ( ℓ j ) − v t,t + r ◦ γ j (0) | + 2 ε = N X j =1 | v t + r ◦ γ j ( ℓ j ) − v t + r ◦ γ j (0) | + 2 ε ≤ pV( v t + r , X ) + 2 ε. Letting ε →
0, we get pV( v t , X ) + pV( v t,t + r , X ) ≤ pV( v t + r , X ). (cid:3) Lemma 3.4.
Let v be a curve-continuous function on X and t ∈ R , r > . Then pV( χ { v>t } , X ) ≤ lim inf r → r pV( v t,t + r , X ) . Proof.
Let γ : [0 , ℓ ] → X be a simple curve. We have for every s ∈ [0 , ℓ ] χ { v>t } ( γ ( s )) = lim r → v t,t + r ( γ ( s )) − tr . In fact, if v ( γ ( s )) ≤ t, then χ { v>t } ( γ ( s )) = 0 and v t,t + r ( γ ( s )) = t . If v ( γ ( s )) > t, then χ { v>t } ( γ ( s )) = 1 . Choose r sufficiently small such that v ( γ ( s )) ≥ t + r for all r ≤ r andthen v t,t + r ( γ ( s )) = t + r .Now | χ { v>t } ◦ γ ( ℓ ) − χ { v>t } ◦ γ (0) | = lim r → r − | v t,t + r ◦ γ ( ℓ ) − v t,t + r ◦ γ (0) | . Let ε >
0. Then take a collection of pairwise disjoint injective curves γ j such thatmin { pV( χ { v>t } , X ) , ε − } ≤ N X j =1 | χ { v>t } ◦ γ j ( ℓ j ) − χ { v>t } ◦ γ j (0) | + ε = N X j =1 lim r → r − | v t,t + r ◦ γ j ( ℓ j ) − v t,t + r ◦ γ j (0) | + ε = lim r → r − N X j =1 | v t,t + r ◦ γ j ( ℓ j ) − v t,t + r ◦ γ j (0) | + ε ≤ lim inf r → r − pV( v t,t + r , X ) + ε. Letting ε →
0, we get the result. (cid:3)
V FUNCTIONS ON ONE-DIMENSIONAL SPACES 14
For any functions v, w on X , we clearly have the subadditivitypV( v + w, X ) ≤ pV( v, X ) + pV( w, X ) . (3.5)Define the inner metric d in by d in ( x, y ) := inf { ℓ γ : γ is a curve such that γ (0) = x, γ ( ℓ γ ) = y } , x, y ∈ X. Denote a ball with respect to the inner metric by B in ( x, r ). Proposition 3.5.
Suppose there exists a constant C such that for all x ∈ X lim inf r → H ( B ( x, r )) r < C holds. Suppose Var( u, X ) < ∞ . Then u is H -measurable and there exists a sequence ofcurve-continuous functions u i → u in L ( X ) such that lim sup i →∞ pV( u i , X ) ≤ C Var( u, X ) . for a constant C that depends only on C .Proof. By Proposition 3.3 we find a good representative v of u . Note that v is necessarilybounded; if it were not, we could fix a point x and find points x j with | v ( x j ) | → ∞ as j → ∞ , and join x to each x j with a curve γ j (by the First Rectifiability Theorem), toget pV( v, X ) ≥ | v ( γ j ( ℓ γ j )) − v ( γ j (0)) | = | v ( x j ) − v ( x ) | → ∞ as j → ∞ . Fix ε >
0. Consider all the points where v is not curve-continuous; such points arecontained in the “jump sets”, defined for κ > J v,κ := { x ∈ X : for all δ > γ j ⊂ B in ( x, δ )such that X j | v ( γ j ( ℓ j )) − v ( γ j (0)) | ≥ κ } . (3.6)We can see that each J v,κ is finite (else we would get pV( v, X ) = ∞ ). Let also J v := S κ> J v,κ . For every x ∈ J v , we define the “size of the jump” J v ( x ) := sup { κ > x ∈ J v,κ } . Let ε >
0. The set J v is at most countable, and so we find an open set W ε ⊃ J v with H ( W ε ) < ε .Let x k be an enumeration of all the points in J v , with the jumps J v ( x k ) in decreasingorder. Note first that by choosing suitable short curves near the jump points, we find thatpV( v, X ) ≥ ∞ X k =1 J v ( x k ) . (3.7) V FUNCTIONS ON ONE-DIMENSIONAL SPACES 15
We modify v as follows. We find r > B = B in ( x , r ) ⊂ W ε and, using also(1.2) (below pV( v, B ) means that all the curves considered are inside 2 B = B in ( x , r ))pV( v, B ) ≤ J v ( x ) and H (2 B ) r < C . (3.8)Choose a function η that is r − -Lipschitz with respect to d in , with η = 1 in B and η = 0outside 2 B . Define ( v B denotes integral average) w := v (1 − η ) + η · v B . Note that J w ⊂ J v \ { x } and that J w ( x k ) ≤ J v ( x k ) for all k ≥ . (3.9)Note also that w = v + η ( v B − v ) and consider pV( η ( v B − v ) , X ). Let γ j be pairwisedisjoint simple curves. Note that η ( v B − v ) = 0 only inside the ball 2 B . By splitting thecurves γ j into subcurves if necessary, we can assume that each of them is contained insidethe ball 2 B . Then we have | ( η ( v B − v ))( γ j ( ℓ j )) − ( η ( v B − v ))( γ j (0)) |≤ | η ( γ j ( ℓ j ))( v B − v )( γ j ( ℓ j )) − η ( γ j ( ℓ j ))( v B − v )( γ j (0)) | + | η ( γ j ( ℓ j ))( v B − v )( γ j (0)) − η ( γ j (0))( v B − v )( γ j (0)) |≤ | v ( γ j ( ℓ j )) − v ( γ j (0)) | + | η ( γ j ( ℓ j )) − η ( γ j (0)) | sup B | v B − v |≤ | v ( γ j ( ℓ j )) − v ( γ j (0)) | + | η ( γ j ( ℓ j )) − η ( γ j (0)) | · J v ( x ) by (3.8) ≤ | v ( γ j ( ℓ j )) − v ( γ j (0)) | + r − ℓ γ j · J v ( x ) . Thus X j | ( η ( v B − v ))( γ j ( ℓ j )) − ( η ( v B − v ))( γ j (0)) |≤ X j | v ( γ j ( ℓ j )) − v ( γ j (0)) | + r − H (2 B )2 J v ( x ) ≤ (2 + 4 C ) J v ( x ) by (3.8)and so pV( η ( v B − v ) , X ) ≤ (2 + 4 C ) J v ( x ) . Finally, by (3.5),pV( w , X ) ≤ pV( v, X ) + pV( η ( v B − v ) , X ) ≤ pV( v, X ) + (2 + 4 C ) J v ( x ) . (3.10)Now we can do this inductively. For each k ∈ N , provided that x k +1 ∈ J w k (if not, wejust let w k +1 = w k ) we choose r k +1 > B k +1 = B in ( x k +1 , r k +1 ) ⊂ W ε andpV( w k , B k +1 ) ≤ J w k ( x k +1 ) and H (2 B k +1 ) r k +1 < C . V FUNCTIONS ON ONE-DIMENSIONAL SPACES 16
As above, we choose a cutoff function η k +1 and then define w k +1 := w k (1 − η k +1 ) + η k +1 · ( w k ) B k +1 . We claim that for all k ∈ N , we havepV( w k , X ) ≤ pV( v, X ) + (2 + 4 C ) k X m =1 J v ( x m )and that J w k ( x m ) ≤ J v ( x m ) for all m ≥ k + 1 . (3.11)We have shown these to be true for k = 1 (recall also (3.9)), and (3.11) is easily seen tohold with k replaced by k + 1. Moreover,pV( w k +1 , X ) ≤ pV( w k , X ) + (2 + 4 C ) J w k ( x k +1 ) (just as in (3.10)) ≤ pV( v, X ) + (2 + 4 C ) k X m =1 J v ( x m ) + (2 + 4 C ) J w k ( x k +1 ) by ind. hyp. ≤ pV( v, X ) + (2 + 4 C ) k +1 X m =1 J v ( x m ) by (3.11) . Then let w := lim k →∞ w k . Note that the convergence is uniform, in particular pointwise,since | w k +1 − w k | ≤ J v ( x k +1 )and recalling (3.7). Now by Proposition 3.4 and (3.7),pV( w, X ) ≤ lim inf k →∞ pV( w k , X ) ≤ pV( v, X ) + (2 + 4 C ) ∞ X k =1 J v ( x k ) ≤ (3 + 4 C ) pV( v, X ) . Since w k has jump discontinuities on curves with jump size at most J v ( x k +1 ) → k → ∞ , and since w k → w uniformly, we see that w is curve-continuous.Recall that w also depends on ε >
0, with w = v outside the open set W ε with H ( W ε ) <ε . Recall also that v is bounded, and furthermore it is easy to check from the constructionthat inf X v ≤ w ≤ sup X v . Choosing ε = 1 /i and letting u i be the corresponding curve-continuous function w , we now get u i → u a.e., and so u is H -measurable by Lemma 3.2,and then u i → u in L ( X ) andlim sup i →∞ pV( u i , X ) ≤ (3 + 4 C ) pV( v, X ) = (3 + 4 C ) Var( u, X ) . (cid:3) V FUNCTIONS ON ONE-DIMENSIONAL SPACES 17
Coarea inequality and the conclusion.
In the last part, we will show a coarea in-equality and prove the implication from sets with finite pointwise variation to finite totalvariation. First we show the following coarea inequality.
Proposition 3.6.
Suppose there exists a constant C such that for all x ∈ X lim inf r → H ( B ( x, r )) r < C holds. Suppose Var( u, X ) < ∞ . Then C Var( u, X ) ≥ ˆ ∗ R Var( χ { u>t } , X ) dt. Note that we use an upper integral since measurability is not clear.
Proof.
First assume that u is curve-continuous and that pV( u, X ) < ∞ . Define (recall that u t = min { t, u } ) m ( t ) := pV( u t , X ) , t ∈ R . Then m is an increasing function and sopV( u, X ) ≥ ˆ ∞−∞ m ′ ( t ) dt. Let ε >
0. Now by Lemma 3.3, m ( t + r ) − m ( t ) ≥ pV( u t,t + r , X ) . Furthermore, Lemma 3.4 implies thatlim inf r → m ( t + r ) − m ( t ) r ≥ lim inf r → pV( u t,t + r , X ) r ≥ pV( χ { u>t } , X ) . Thus we have pV( u, X ) ≥ ˆ ∗ R pV( χ { u>t } , X ) dt ≥ ˆ ∗ R Var( χ { u>t } , X ) dt. (3.12)Now for a general function u on X with Var( u, X ) < ∞ , by Proposition 3.5 we find asequence of curve-continuous functions u i with u i → u in L ( X ) andlim sup i →∞ pV( u i , X ) ≤ C Var( u, X ) . For every x ∈ X , ˆ ∞−∞ | χ { u i >t } ( x ) − χ { u>t } ( x ) | dt = ˆ max { u i ( x ) ,u ( x ) } min { u i ( x ) ,u ( x ) } dt = | u i ( x ) − u ( x ) | . V FUNCTIONS ON ONE-DIMENSIONAL SPACES 18
Hence by Fubini’s theorem (recall the measurability statement of Proposition 3.5) ˆ X | u i − u | d H = ˆ X ˆ ∞−∞ | χ { u i >t } ( x ) − χ { u>t } ( x ) | dt d H ( x )= ˆ ∞−∞ ˆ X | χ { u i >t } ( x ) − χ { u>t } ( x ) | d H ( x ) dt. Thus k χ { u i >t } − χ { u>t } k L ( X ) → L ( R ) and so we can find a subsequence of u i (notrelabeled) such that k χ { u i >t } − χ { u>t } k L ( X ) → t ∈ R . Then for such t , by the lower semicontinuity of Proposition 3.4,Var( χ { u>t } , X ) ≤ lim inf i →∞ Var( χ { u i >t } , X ) . We find measurable functions h i ≥ χ { u i >t } on R such thatlim inf i →∞ ˆ ∞−∞ h i ( t ) dt = lim inf i →∞ ˆ ∗ R Var( χ { u i >t } , X ) dt. Then by Fatou’s lemma ˆ ∗ R Var( χ { u>t } , X ) dt ≤ ˆ ∗ R lim inf i →∞ Var( χ { u i >t } , X ) dt ≤ ˆ R lim inf i →∞ h i ( t ) dt ≤ lim inf i →∞ ˆ R h i ( t ) dt = lim inf i →∞ ˆ ∗ R Var( χ { u i >t } , X ) dt ≤ lim inf i →∞ pV( u i , X ) by (3.12) ≤ C Var( u, X ) . (cid:3) Due to the above coarea inequality, it will suffice to consider characteristic functions u = χ E for E ⊂ X . Proposition 3.7.
Suppose there exists a constant C such that for all x ∈ X lim inf r → H ( B ( x, r )) r < C holds. Let E ⊂ X . Then k Dχ E k ( X ) ≤ C Var( χ E , X ) .Proof. We can assume that Var( χ E , X ) < ∞ . By Proposition 3.3 we find a good represen-tative v of χ E , so that pV( v, X ) = Var( χ E , X ). Let D := { x ∈ X : v ( x ) ∈ { , }} , so that H ( X \ D ) = 0. By Proposition 3.2 and its proof, we know that there is a function v e on V FUNCTIONS ON ONE-DIMENSIONAL SPACES 19 X with v e = v on D , taking only the values 0 ,
1, with pV( v e , X ) = pV D ( v, X ) ≤ pV( v, X )and so in fact pV( v e , X ) = Var( χ E , X ). In conclusion, we can take the good representativeto be χ F for F ⊂ X , and then pV( χ F , X ) = Var( χ E , X ).Recall the definition of the jump set from (3.6); it is not difficult to see that now J χ F = { x ∈ X : for all δ > γ ⊂ B in ( x, δ )that intersects both F and X \ F } . We call this the “curve boundary” ∂ c F := J χ F . Clearly any curve intersecting both F and X \ F needs to intersect also ∂ c F . Now if H ( ∂ c F ) = ∞ , then we can pick arbitrarily manydisjoint curves γ : [0 , ℓ ] → X with | χ F ( γ ( ℓ )) − χ F ( γ (0)) | = 1 and thus pV( χ F , X ) = ∞ .But since pV( χ F , X ) < ∞ , actually H ( ∂ c F ) < ∞ . In other words, ∂ c F = { x , . . . , x N } with N ≤ pV( χ F , X ) = Var( χ E , X ).Take a sequence δ i ց B ( x j , δ ), j = 1 , . . . , N , are pairwise disjoint.Fix i ∈ N . By (1.2), for each j = 1 , . . . , N we find δ j,i ∈ (0 , δ i ) such that H ( B ( x j , δ j,i )) δ j,i < C . (3.13)For each j = 1 , . . . , N , let η j,i be a 1 /δ j,i -Lipschitz function with η j,i ( x j ) = 1 and η j,i = 0outside B ( x j , δ j,i ). Define v i := max { η ,i , . . . , η N,i } and u i := max { χ F , η ,i , . . . , η N,i } . Let g i := N X j =1 χ B ( x j ,δ j,i ) δ j,i . Note that since the pointwise Lipschitz constant (2.3) is an upper gradient [5, Proposition1.14], and by [5, Corollary 2.21], we know that χ B ( x j ,δ j,i ) /δ j,i is a 1-weak upper gradient of η j,i (recall Definition 2.1). Then g i is a 1-weak upper gradient of v i .Then we can verify that g i is a 1-weak upper gradient of u i . For this we need to checkthree cases for a curve γ : [0 , ℓ ] → X with end points γ (0) = x and γ ( ℓ ) = y . We canassume that the pair ( v i , g i ) satisfies the upper gradient inequality on the curve γ as wellas all of its subcurves [5, Lemma 1.40]. The first case is x, y ∈ F , where | u i ( x ) − u i ( y ) | = 0 ≤ ˆ γ g i ds. The second case is x, y ∈ X \ F . Here | u i ( x ) − u i ( y ) | = | v i ( x ) − v i ( y ) | ≤ ˆ γ g i ds. The third case is x ∈ F and y ∈ X \ F . As mentioned before, γ now necessarily intersects ∂ c F . Thus there is some t ∈ [0 , ℓ ] such that γ ( t ) ∈ ∂ c F , and thus γ ( t ) = x j for some j . V FUNCTIONS ON ONE-DIMENSIONAL SPACES 20
Note that u i ( γ (0)) = 1, u i ( γ ( t )) = v i ( γ ( t )) = 1, and u i ( γ ( ℓ )) = v i ( γ ( ℓ )). It follows that | u i ( γ ( ℓ )) − u i ( γ (0)) | ≤ | u i ( γ ( ℓ )) − u i ( γ ( t )) | + | u i ( γ ( t )) − u i ( γ (0)) | = | v i ( γ ( ℓ )) − v i ( γ ( t )) | ≤ ˆ γ g i ds. In conclusion, g i is a 1-weak upper gradient of u i . It is easy to see that also u i → χ E in L ( X ). Now we have, using (3.13), k Dχ E k ( X ) ≤ lim inf i →∞ ˆ X g i d H ≤ lim inf i →∞ N X j =1 H ( B ( x j , δ j,i )) δ j,i ≤ C N ≤ C Var( χ E , X ) . (cid:3) Proposition 3.8.
Suppose there exists a constant C such that for all x ∈ X lim inf r → H ( B ( x, r )) r < C holds. Suppose Var( u, X ) < ∞ . Then k Du k ( X ) ≤ C Var( u, X ) .Proof. From Var( u, X ) < ∞ it follows that u is essentially bounded, and u is H -measurableby Proposition 3.5. Combined with the fact that H ( X ) < ∞ , we get u ∈ L ( X ). By theBV coarea formula (2.4), Proposition 3.7, and the coarea inequality of Proposition 3.6, itfollows that k Du k ( X ) = ˆ ∗ R k Dχ { u>t } k ( X ) dt ≤ C ˆ ∗ R Var( χ { u>t } , X ) dt ≤ C C Var( u, X ) . (cid:3) Theorem 1.1 follows by combining Proposition 3.1 and Proposition 3.8.4.
Federer’s characterization of sets of finite perimeter
Let us briefly consider a more general metric space (
X, d, µ ), where µ is a Radon measure.The codimension one Hausdorff measure is defined for any set A ⊂ X by H ( A ) := lim R → H R ( A )with H R ( A ) := inf (X i ∈ I µ ( B ( x i , r i )) r i : A ⊂ [ i ∈ I B ( x i , r i ) , r i ≤ R ) , where I ⊂ N is a finite or countable index set. Note that in an Ahlfors one-regular space, H is comparable to H .Given any set E ⊂ X , the measure-theoretic boundary ∂ ∗ E is the set of points x ∈ X for which lim sup r → µ ( B ( x, r ) ∩ E ) µ ( B ( x, r )) > r → µ ( B ( x, r ) \ E ) µ ( B ( x, r )) > . V FUNCTIONS ON ONE-DIMENSIONAL SPACES 21
Recall from the Introduction that if (
X, d, µ ) is a complete metric space such that µ isdoubling and the space supports a 1-Poincar´e inequality, then the condition H ( ∂ ∗ E ) < ∞ for a measurable set E ⊂ X implies that k Dχ E k ( X ) < ∞ . This is the “if” direction ofFederer’s characterization of sets of finite perimeter.Define a space as a subset of R as follows. First define for each j ∈ N A j := j − [ k =0 I jk , where I jk := (cid:26)(cid:18) t cos (cid:18) kπ j (cid:19) , t sin (cid:18) kπ j (cid:19)(cid:19) ∈ R : t ∈ [ − , (cid:27) is a line segment passing through the origin with length H ( I jk ) = 2. The angle between I jk and the positive x -axis is kπ j and the angle between I jk and I jk − is π j . For any set A ⊂ R and a >
0, we let aA := { ( ax, ay ) : ( x, y ) ∈ A } . Then consider e A j := 2 − j − A j for each j ∈ N . Note that e A j is a collection of 2 j linesegments e I jk with length H ( e I jk ) = 2 − j .Define X := ∞ [ j =1 e A j . (4.1)We first show that the doubling condition is essential in the “if” direction of Federer’scharacterization. Example 4.1.
Equip the set X in (4.1) with the geodesic metric and the measure H . Wehave H ( X ) ≤ ∞ X j =1 j H ( e I jk ) = ∞ X j =1 − j = 1 . Clearly, the density upper bound condition (1.2) no longer holds at 0. Moreover, H is notdoubling: the doubling condition fails when we choose points x close to 0 with 0 ∈ B ( x, r )and 0 / ∈ B ( x, r ).Now we show that this space does support a 1-Poincar´e inequality. First consider a ball B (0 , r ). Suppose u is a function on X with u (0) = 0 and let g be an upper gradient of u .Every x ∈ B (0 , r ) is connected to 0 by a line segment I . We have ˆ I g d H ≥ | u ( x ) − u (0) | = | u ( x ) | . Note that B (0 , r ) consists of countably many line segments { I j } ∞ j =1 that have the origin asone end point (some may be half-open). By the above, we have | u ( x ) | ≤ ˆ I j g d H for every x ∈ I j . V FUNCTIONS ON ONE-DIMENSIONAL SPACES 22
Thus ˆ B (0 ,r ) | u | d H = ∞ X j =1 ˆ I j | u | d H ≤ ∞ X j =1 (cid:16) H ( I j ) ˆ I j g d H (cid:17) ≤ r ˆ B (0 ,r ) g d H since H ( I j ) ≤ r for all j ∈ N . Now consider a general ball B ( x, r ) and a function u ∈ L ( X ) with upper gradient g . If B ( x, r ) is contained in only one line segment, the Poincar´e inequality obviouslyholds since it holds in R . So we can assume that 0 ∈ B ( x, r ). We can also assume that ´ B (0 , r ) g d H < ∞ and then u is a bounded function in B (0 , r ). Thus we can assume that u (0) = 0. Now ˆ B ( x,r ) | u − u B ( x,r ) | d H ≤ ˆ B ( x,r ) | u | d H (see e.g. [5, Lemma 4.17]) ≤ ˆ B (0 , r ) | u | d H ≤ r ˆ B (0 , r ) g d H ≤ r ˆ B ( x, r ) g d H . Thus a 1-Poincar´e inequality holds with C P = 4 and λ = 3.Next, for each j ∈ N choose I j = { ( t cos(2 − j π ) , t sin(2 − j π )) , t ∈ [ − , } and then let E := ∞ [ j =1 e I j = ∞ [ j =1 − j − I j . (4.2)Consider any sequence ( u i ) ⊂ N , ( X ) with u i → χ E in L ( X ), with upper gradients g i .We can also assume that u i → χ E a.e. Thus for each j ∈ N we can choose a point x j ∈ e I j , x j = 0 and a point x ′ j in e A j \ e I j such that(1) u i ( x j ) → i → ∞ ;(2) u i ( x ′ j ) → i → ∞ ;(3) the curves γ j joining x ′ j and x j only intersect at the origin.Now ˆ X g i d H ≥ ∞ X j =1 ˆ γ j g i d H ≥ ∞ X j =1 | u i ( x ′ j ) − u i ( x j ) | → ∞ as i → ∞ . Hence k Dχ E k ( X ) = ∞ .It is easy to check that 0 / ∈ ∂ ∗ E and then in fact ∂ ∗ E = ∅ . This shows that the “if”direction of Federer’s characterization does not hold without the doubling condition. V FUNCTIONS ON ONE-DIMENSIONAL SPACES 23
On the other hand, pV( χ E , X ) = 1 since only a curve intersecting 0 can give nonzerovariation. Thus we do need condition (1.2) in Proposition 3.7 and Proposition 3.8.The following example shows that the Poincar´e inequality cannot be dropped in theimplication from H ( ∂ ∗ E ) < ∞ to k Dχ E k ( X ) < ∞ either. Example 4.2.
Equip the set X in (4.1) with the metric inherited from R and the mea-sure H . In this case, we will show that H is doubling on X , but X does not supportany Poincar´e inequality since it is clearly not quasiconvex (recall Definition 2.2 and theparagraph after it). Let x ∈ X . If x = 0, we have 2 − k − ≤ d ( x, ≤ − k − for some k ∈ N . Suppose first that r ≤ − k − . Recalling the notation from the previous example,note that e A k consists of 2 k line segments, which are at angles 2 π × − k − from each other.By simple geometric reasoning we see that the ball B ( x, r/
2) is intersected by at least r × k − × (2 π × − k − ) − ≥ k − r line segments belonging to e A k , each for a length at least r/ B ( x, r ). Thus H ( B ( x, r )) ≥ k − r . To prove a converse estimate, suppose still that 2 − k − ≤ d ( x, ≤ − k − , and supposethat 2 − k − ≤ r ≤ − k − . We have B ( x, r ) ∩ e A j = ∅ for all j ≥ k + 2. Note that e A k +1 consists of 2 k +1 line segments, which are at angles 2 π × − k − from each other. Thus wecan see that there are at most4 r × k +4 × (2 π × − k − ) − ≤ k +6 r line segments intersecting B ( x, r ), each for a length at most 2 r . Thus H ( B ( x, r )) ≤ k +7 r . Thus in total 2 k − r ≤ H ( B ( x, r )) ≤ k +7 r , (4.3)where the first inequality holds for all r ≤ − k − and the second for all 2 − k − ≤ r ≤ − k − .Moreover, for every k ∈ N , H ( B (0 , − k − )) ≥ − k − H ( A k ) = 2 − k − k +1 = 2 − k and so2 − k ≤ H ( B (0 , − k − )) ≤ ∞ X j = k − j − H ( A j ) = ∞ X j = k − j − j +1 = 2 − k +1 . (4.4)From these, the doubling condition for balls centered at 0 easily follows. Now assume againthat x = 0, so that 2 − k − ≤ d ( x, ≤ − k − for a given k ∈ N . We consider four cases: V FUNCTIONS ON ONE-DIMENSIONAL SPACES 24 (1) If
R < − k − , then B ( x, R ) consists of just one line segment and so H ( B ( x, R )) = 2 H ( B ( x, R )) . (2) If 2 − k − ≤ R ≤ − k − , then by (4.3),2 k − R ≤ H ( B ( x, R )) and H ( B ( x, R )) ≤ k +7 (2 R ) , and so we have H ( B ( x, R )) ≤ H ( B ( x, R )) . (3) If 2 − k − < R ≤ − k +1 , then applying (4.3) with r = 2 − k − , H ( B ( x, R )) ≥ H ( B ( x, − k − )) ≥ k − (2 − k − ) = 2 − k − and by (4.4), H ( B ( x, R )) ≤ H ( B (0 , − k +2 )) ≤ − k +3 , and so we have H ( B ( x, R )) ≤ H ( B ( x, R )) . (4) If 2 − k +1 < R ≤ − with k ≥ X = 2 − ), we choose j ≤ k such that2 − j +1 < R ≤ − j +3 . Note that B (0 , R/ ⊂ B ( x, R ) ⊂ B ( x, R ) ⊂ B (0 , R ). Nowby (4.4), H ( B ( x, R )) ≥ H ( B (0 , R/ ≥ H ( B (0 , − j )) ≥ − j and H ( B ( x, R )) ≤ H ( B (0 , R )) ≤ H ( B (0 , − j +5 )) ≤ − j +4 . Thus H ( B ( x, R )) ≤ H ( B ( x, R )) . In total, the doubling condition always holds with doubling constant 2 , when x = 0.Finally, define the set E as in (4.2). As before, we obtain that k Dχ E k ( X ) = ∞ ,pV( χ E , X ) = 1, and ∂ ∗ E = ∅ . Thus again we see that the “if” direction of Federer’scharacterization does not hold, and that condition (1.2) is needed in Proposition 3.7 andProposition 3.8. References [1] L. Ambrosio,
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Panu Lahti, Institut f¨ur Mathematik, Universit¨at Augsburg, Universit¨atsstr. 14, 86159Augsburg, Germany, [email protected]