POINTWISE PROPERTIES OF FUNCTIONS OF BOUNDED ...

POINTWISE PROPERTIES OF FUNCTIONS OF BOUNDED VARIATION IN METRIC SPACES JUHA KINNUNEN, RIIKKA KORTE, NAGESWARI SHANMUGALINGAM AND HELI TUOMINEN Abstract. We study pointwise properties of functions of bounded variation on a metric space equipped with a doubling measure and supporting a Poincaré inequality. In particular, we obtain a Lebesgue type result for BV functions. We also study approximations by Lipschitz continuous functions and a version of the Leibniz rule. We give examples which show that our results are optimal for BV functions in this generality.

1. Introduction This paper studies Lebesgue points for functions of bounded variation on a metric measure space (X, d, µ) equipped with a doubling measure and supporting a Poincáre inequality. Here the metric d and the measure µ will be fixed, and we denote the triple (X, d, µ) simply by X. We say that x ∈ X is a Lebesgue point of a locally integrable function u if the limit Z u dµ

lim

r→0

B(x,r)

exists. Observe that if almost every point is a Lebesgue point of every function u in a certain class of functions, then Z lim |u − u(x)| dµ r→0 B(x,r)

for almost every x ∈ X. This follows easily by considering |u − q| with q ∈ Q. The set of non-Lebesgue points of a classical Sobolev function is a set of measure zero with respect the Hausdorff measure of codimension one and this holds true also in metric spaces supporting a doubling measure and a Poincaré inequality, see [11] and [12]. The situation is more delicate for BV functions. It is known that in the Euclidean case with Lebesgue measure, a BV function has Lebesgue points outside a null set of Hausdorff measure of codimension one. More 1991 Mathematics Subject Classification. 26A45, 30L99. Part of this research was conducted during the visit of the third author to Aalto University; she wishes to thank that institution for its kind hospitality. The research is supported by the Academy of Finland. 1

2

KINNUNEN, KORTE, SHANMUGALINGAM AND TUOMINEN

precisely, Z lim

(1.1)

r→0 B(x,r)

u dy =

1 ∧ u (x) + u∨ (x) 2

for Hn−1 -almost every x, see [6, Corollary 1 of page 216], [7, Theorem 4.5.9] and [20, Theorem 5.14.4]. Here u∧ (x) and u∨ (x) are the lower and upper approximate limits of u at x, respectively. The proof of this result lies rather deep in the theory of BV functions and it seems to be very sensitive to the measure. Indeed, we give a simple example which shows that the corresponding result is not true even in the Euclidean case with a weighted measure. However, we are able to show the following metric space analogue of the result. The Hausdorff measure of codimension one is denoted by H. The precise definitions will be given in Section 2. Theorem 1.1. Assume that µ is doubling and that X supports a weak (1, 1)-Poincaré inequality. If u ∈ BV (X), then for H-almost every x ∈ X, we have Z ∧ ∨ (1 − γ)u (x) + γu (x) ≤ lim inf u dµ r→0 B(x,r) Z u dµ ≤ γu∧ (x) + (1 − γ)u∨ (x), ≤ lim sup r→0

B(x,r)

1 2

where 0 < γ ≤ and γ depends only on the doubling constant and the constants in the in the weak (1, 1)-Poincaré inequality. We also give an example which shows that, unlike in the classical Euclidean setting, we cannot hope to get the existence of the Lebesgue points H-almost everywhere in X. However, the main result above indicates that both limes infimum and limes supremum are comparable to the pointwise representative u defined by Z u dµ (1.2) u(x) = lim r→0 B(x,r)

for H-almost every in x ∈ X. As an application of Theorem 1.1, we study approximations of BV functions. By definition, a BV function can be approximated by locally Lipschitz continuous functions in L1 (X) so that the integral of upper gradients converges to the total variation measure. In some applications, a better control on pointwise convergence would be desirable. We construct two approximation results and apply one of the approximations in proving a version of the Leibniz rule for bounded BV functions. In the Euclidean case, the corresponding result has been studied in [5], [18] and [19]. An unexpected feature is that a multiplicative constant appears, which is not present for Sobolev functions and we show that this result is optimal.

POINTWISE PROPERTIES OF BV FUNCTIONS OIN METRIC SPACES

3

2. Preliminaries In this paper, (X, d, µ) is a complete metric measure space with a Borel regular outer measure µ. The measure is assumed to be doubling. This means that there exists a constant cD > 0 such that for all x ∈ X and r > 0, µ(B(x, 2r)) ≤ cD µ(B(x, r)). It follows from the doubling condition that Q µ(B(y, R)) R ≤c µ(B(x, r)) r for every r ≤ R and y ∈ B(x, r) for some Q > 1 that only depends on cD . We recall that complete metric space endowed with a doubling measure is proper, that is, closed and bounded sets are compact. A nonnegative Borel function g on X is an upper gradient of an extended real valued function u on X if for all paths γ in X we have Z (2.1) |u(x) − u(y)| ≤ g ds, γ

R whenever both u(x) and u(y) are finite, and γ g ds = ∞ otherwise. Here x and y are the end points of γ. If g is a nonnegative measurable function on X and if (2.1) holds for almost every path with respect to the 1-modulus, then g is a 1-weak upper gradient of u. The collection of all upper gradients, together, play the role of the modulus of the weak derivative of a Sobolev function in the metric setting. We consider the following norm kf kN 1,1 (X) := kf kL1 (X) + inf kgkL1 (X) g

with the infimum taken over all upper gradients g of f . The NewtonSobolev space considered in this note is the space N 1,1 (X) = {u : kukN 1,1 (X) < ∞}/∼, where the equivalence relation ∼ is given by f ∼ h if and only if kf − hkN 1,1 (X) = 0. Next we recall the definition and basic properties of functions of bounded variation on metric spaces, see [14]. Definition 2.1. For u ∈ L1loc (X), we define kDuk(X) Z n o 1 = inf lim inf gui dµ : ui ∈ Liploc (X), ui → u in Lloc (X) , i→∞

X

where gui is a 1-weak upper gradient of ui . We say that a function u ∈ L1 (X) is of bounded variation, u ∈ BV (X), if kDuk(X) < ∞. Moreover, a measurable set E ⊂ X is said to have finite perimeter if

4


kDχE k(X) < ∞. By replacing X with an open set U ⊂ X, we may define kDuk(U ) and we denote P (E, U ) = kDχE k(U ), the perimeter of E in U . By [14, Theorem 3.4], we have the following result. Theorem 2.2. Let u ∈ BV (X). For a set A ⊂ X, we define kDuk(A) = inf kDuk(U ) : U ⊃ A, U ⊂ X is open . Then kDuk(·) is a finite Borel outer measure. Let E be a set of finite perimeter in X. For every set A ⊂ X, we denote P (E, A) = kDχE k(A). We say that X supports a weak (1, 1)-Poincaré inequality if there exist constants cP > 0 and τ > 1 such that for all balls B = B(x, r), all locally integrable functions u and for all 1-weak upper gradients g of u, we have Z Z (2.2) |u − uB | dµ ≤ cP r g dµ, B

where

τB

Z 1 uB = u dµ = u dµ. µ(B) B B If the space supports a weak (1, 1)-Poincaré inequality, then for every u ∈ BV (X), we have Z kDuk(τ B) (2.3) |u − uB | dµ ≤ cP r , µ(B) B where the constant cP and the dilation factor τ are the same constants as in (2.2). The (1, 1)-Poincaré inequality implies that Z (Q−1)/Q kDuk(τ B) (2.4) |u − uB |Q/(Q−1) dµ ≤ cr , µ(B) B Z

see [9, Theorem 9.7]. Here the constant c > 0 depends only on the doubling constant and the constants in the Poincaré inequality. We assume, without further notice, that the measure µ is doubling and that the space supports a weak (1, 1)-Poincaré inequality The following coarea formula holds for BV functions. For a proof see [14, Proposition 4.2]. If u ∈ L1loc (X) and A ⊂ X is open, then Z ∞ (2.5) kDuk(A) = P ({u > t}, A) dt. −∞

In particular, if u ∈ BV (X), then the set {u > t} has finite perimeter for almost every t ∈ R and formula (2.5) holds for all Borel sets A ⊂ X.


5

The restricted spherical Hausdorff content of codimension one on X is HR (E) = inf

∞ nX µ(B(x , r )) i

i=1

i

:E⊂

ri

∞ [

o B(xi , ri ), ri ≤ R ,

i=1

where 0 < R < ∞. When R = ∞, the infimum is taken over coverings with finite radius. The number H∞ (E) is the Hausdorff content of E. The Hausdorff measure of codimension one of E ⊂ X is H(E) = lim HR (E). R→0

A combination of [3, Theorems 4.4 and 4.6] gives the equivalence of the perimeter measure and the Hausdorff measure of codimension one for sets of finite perimeter. The measure theoretic boundary of E, denoted by ∂ ∗ E, is the set of points x ∈ X, where both E and its complement have positive density, i.e. µ(E ∩ B(x, r)) µ(B(x, r) \ E) lim sup > 0 and lim sup > 0. µ(B(x, r)) µ(B(x, r)) r→0 r→0 The following result is extremely useful for us. For the proof, we refer to [1] and [3]. Theorem 2.3. Assume that E is a set of finite perimeter. Then 1 (2.6) P (E, A) ≤ H(∂ ∗ E ∩ A) ≤ cP (E, A), c where c depends only on the doubling constant and the constants in the Poincaré inequality. Moreover, the following theorem shows that we can consider even a smaller part of the measure theoretic boundary. For a proof, see [1, Theorem 5.3]. Theorem 2.4. Let E be a set of finite perimeter. For γ > 0, we define Σγ (E) to be the set consisting of all points x ∈ X for which n µ(E ∩ B(x, r)) µ(B(x, r) \ E) o lim inf min , ≥ γ. r→0 µ(B(x, r)) µ(B(x, r)) Then there exists γ > 0, depending only on the doubling constant and the constants in the Poincaré inequality, such that H(∂ ∗ E \ Σγ (E)) = 0. Let E ⊂ X be a Borel set. The upper density of E at a point x ∈ X is D(E, x) = lim sup r→0

µ(B(x, r) ∩ E) µ(B(x, r))

and the lower density D(E, x) = lim inf r→0

µ(B(x, r) ∩ E) . µ(B(x, r))

6


If D(E, x) = D(E, x) then the limit exists and we denote it by D(E, x). By the differentiation theory for doubling measures, we have D(E, x) = 1 for µ-almost every x ∈ E and D(E, x) = 0 for µ-almost every x ∈ X \ E. See, for example, the discussion in [10, Section 2.7] or [15]. Following the notation of [3], we define upper and lower approximate limits. Definition 2.5. Let u : X → [−∞, ∞] be a measurable function. The upper and lower approximate limit of u at x ∈ X are u∨ (x) = inf t : D({u > t}, x) = 0 and u∧ (x) = sup t : D({u < t}, x) = 0 . If u∨ (x) = u∧ (x), then the common value is denoted by u e(x) and called the approximate limit of u at x. The function u is approximately continuous at x if u e(x) = u(x). The approximate jump set of u is Su = {u∧ < u∨ }. By the Lebesgue differentiation theorem, a locally integrable function u is approximately continuous at µ-almost every point and hence µ(Su ) = 0. A similar argument as in the classical case of [20, Remark 5.9.2] shows that a function u is approximately continuous at x if and only if there exists a measurable set E such that x ∈ E, D(E, x) = 1 and the restriction of u to E is continuous. We need the following standard measure theoretic lemma. We recall the proof here to emphasize the fact that it also applies in the metric context. Lemma 2.6. Let u ∈ BV (X). Let A be a Borel set such that lim sup r r→0

kDuk(B(x, r)) >λ µ(B(x, r))

for all x ∈ A. Then there is a constant c > 0, depending only on the doubling constant, such that kDuk(A) ≥ cλH(A). Proof. Let ε > 0. Let U be an open set such that A ⊂ U . For each x ∈ A there exists 0 < rx ≤ min{ε, dist(x, X \ U )}/5 such that (2.7)

µ(B(x, rx )) kDuk(B(x, rx )) < . rx λ

By a general covering theorem, there is a subfamily of disjoint balls Bi = B(xi , ri ) ⊂ U , i = 1, 2, . . . , such that (2.7) holds for each Bi and


7

A ⊂ ∪∞ i=1 5Bi . Using the doubling condition, (2.7) and the pairwise disjointedness of the balls Bi , we obtain Hε (A) ≤

∞ X µ(5Bi )

5ri

i=1

≤c

∞ X µ(Bi ) i=1

ri

∞

cX c ≤ kDuk(Bi ) ≤ kDuk(U ). λ i=1 λ Since kDuk(A) = inf{kDuk(U ) : U is open, A ⊂ U }, the claim follows by letting ε → 0 and then taking the infimum over open sets U . 3. Lebesgue points In this section, we use the approximately continuous representative u e of u, and denote it by u. Our first lemma shows that the Poincaré inequality holds for small radii without the integral average on the left-hand side if the approximate limit of the function is zero in the center of the ball. Lemma 3.1. Let u ∈ BV (X) and let x0 ∈ X. If u(x0 ) = 0, then Z (Q−1)/Q kDuk(B(x0 , r)) lim sup |u|Q/(Q−1) dµ ≤ c lim sup r . µ(B(x0 , r)) r→0 r→0 B(x0 ,r) Here the constant c > 0 depends only on the doubling constant and the constants in the Poincaré inequality. Proof. Let ε > 0 and p = Q/(Q − 1). Since 0 is the approximate limit of u at x0 , we have D({|u| > t}, x0 ) = 0 for all t > 0. Thus there exists rε > 0 such that µ({|u| > t} ∩ B(x0 , r)) < εµ(B(x0 , r)) whenever r ≤ rε . This implies that for sets B = B(x0 , r) and E = {|u| ≤ t} ∩ B, we have µ(E) ≥ (1 − ε)µ(B). From this together with the Sobolev-Poincaré inequality (2.4) we conclude that Z Z 1/p 1/p µ(B) p |u − uE | dµ ≤2 |u − uB |p dµ µ(E) B B 2 kDuk(τ B) ≤ cr . 1−ε µ(B)

8


Hence Z

1/p Z 1/p |u| dµ |u − uE |p dµ ≤ + |u|E p

B

B

kDuk(τ B) 2 cr +t ≤ 1−ε µ(B) for 0 < r ≤ rε . The claim follows by letting t → 0 and then taking the limes superior on both sides as r → 0. Lemma 3.2. Let u ∈ BV (X). Then −∞ < u∧ (x) ≤ u∨ (x) < ∞ for H-almost every x ∈ X. Proof. Since the question is of local nature, without loss of generality, we may assume that u has compact support. First we will show that H({u∧ = ∞}) = 0. For t ∈ R, let Ft = {u∧ > t} and Et = {u > t}. The definitions of u∧ and Ft imply that D(Et , x) = 1 for each x ∈ Ft . Since µ-almost every point is a Lebesgue point of u, we have u∧ (x) = u(x) for µ-almost every x and therefore D(Ft , x) = D(Et , x) = 1 when x is a Lebesgue point of u. The boxing inequality ([12, Remark 3.3.(1)] and [16]) gives disjoint balls Bi = B(xi , τ ri ), i = 1, 2, . . . , such that Ft ⊂ ∪∞ i=1 5Bi and ∞ X µ(5Bi )

5τ ri

i=1

≤ cP (Ft , X).

Since u has compact support, we have ri ≤ R, i = 1, 2, . . ., for R > 0 large enough. This implies that HR (Ft ) ≤ cP (Ft , X) and \ ∧ HR ({u = ∞}) = HR Ft ≤ lim inf HR (Ft ) t→∞ t>0 (3.1) ≤ lim inf cP (Ft , X). t→∞

∧

As u = u µ-almost everywhere, we also have P (Et , X) = P (Ft , X) and, by the coarea formula, Z ∞ Z P (Ft , X) dt = −∞

∞

P (Et , X) dt = kDuk(X) < ∞.

−∞

From this we conclude that lim inf P (Ft , X) = 0. t→∞


9

By (3.1) we have HR ({u∧ = ∞}) = 0 and consequently also H({u∧ = ∞}) = 0, see for example the proof of [12, Lemma 7.6]. A similar argument shows that H({u∨ = −∞}) = 0. The first part of the proof shows that u∨ −u∧ is well defined H-almost everywhere. Next we show that H({u∨ − u∧ = ∞}) = 0. It follows from the definitions of the approximate limits and the measure theoretic boundary that At = {u∧ < t < u∨ } ⊂ ∂ ∗ Et for every t ∈ R. Then, by the Fubini theorem, Z ∞ Z ∞Z χAt (x) dH(x) dt H(At ) dt = −∞ −∞ X Z Z ∞ χAt (x) dt dH(x) = X

(3.2)

−∞ u∨ (x)

Z Z =

1 dt dH(x) X

Z =

u∧ (x)

(u∨ (x) − u∧ (x)) dH(x).

X

By (3.2), Theorem 2.3, and the coarea formula, we have Z ∞ Z ∨ ∧ H({u∧ < t < u∨ }) dt (u − u ) dH ≤ Su Z−∞ ∞ ≤ H(∂ ∗ Et ) dt −∞ Z ∞ ≤c P (Et , X) dt −∞

= ckDuk(X) < ∞. Since u∧ = u∨ outside Su , the claim follows from this.

The next example shows that (1.1) does not even hold for BV functions in weighted Euclidean spaces. Example 3.3. Let X = R2 with the Euclidean distance and the measure with the density dµ = ωdL2 , where ω = 1+χB(0,1) . Let u = χB(0,1) . Then for every x ∈ ∂B(0, 1), we have Z 1 ∧ 1 2 ∨ (u (x) + u (x)) = 6= = lim u dµ. 2 2 3 r→0 B(x,r)

10


The above example also shows that we cannot always take γ = 1/2 in Theorem 1.1, for in this example we have γ = 1/3. We will start the proof of our main result, Theorem 1.1, by showing that the claim of Theorem 1.1 holds outside the approximate jump set if u is bounded. Later we remove the extra assumption on boundedness by a limiting argument. Lemma 3.4. Let u ∈ BV (X) ∩ L∞ (X). Then Z lim |u − u(x0 )|Q/(Q−1) dµ = 0. r→0 B(x ,r) 0

for all x0 ∈ X \ Su . Proof. Let x0 ∈ X \ Su . Since we use the representative u e and u is approximately continuous at x0 , there is a measurable set A such that x0 ∈ A, D(A, x0 ) = 1 and lim

x→x0 ,x∈A

u(x) = u(x0 ).

Let r > 0, B = B(x0 , r) and p = Q/(Q − 1). Then we have Z Z 1 p |u − u(x0 )| dµ = |u − u(x0 )|p dµ µ(B) B B∩A Z (3.3) 1 + |u − u(x0 )|p dµ. µ(B) B\A The second term on the right hand side of (3.3) has an upper bound Z 1 µ(B \ A) |u − u(x0 )|p dµ ≤ 2kukp∞ , µ(B) B\A µ(B) which tends to zero as r → 0 because D(X \ A, x0 ) = 0. Then we estimate the first term on the right hand side of (3.3). Let 0 < ε < 1. There is rε > 0 such that |u(x) − u(x0 )| < ε when d(x, x0 ) < rε and x ∈ A. Hence, we obtain Z 1 µ(B ∩ A) |u − u(x0 )|p dµ ≤ εp ≤ εp µ(B) B∩A µ(B) for r < rε . The claim follows by letting ε → 0.

Now we generalize the previous lemma for unbounded BV functions. The proof is similar to the Euclidean argument of [20, Theorem 4.14.3]. Theorem 3.5. Let u ∈ BV (X). Then Z lim |u − u(x0 )|Q/(Q−1) dµ = 0. r→0 B(x ,r) 0

for H-almost all x0 ∈ X \ Su .


11

Proof. Let 0 < ε < 1 and p = Q/(Q − 1) and denote Wi = {−i ≤ u∧ ≤ u∨ ≤ i},

i = 1, 2, . . .

By Lemma 3.2, we have

H X\

∞ [

Wi = 0,

i=1

S and hence it is enough to prove the claim for x0 ∈ ( ∞ i=1 Wi ) \ Su . For i = 1, 2, . . . , let ui be a truncation of u defined as   if u(x) > i, i, ui (x) = u(x), if |u(x)| ≤ i,  −i, if u(x) < −i. By the Minkowski inequality, we have Z 1/p |u − u(x0 )|p dµ B(x0 ,r) (3.4) Z 1/p Z p ≤ |u − ui | dµ + B(x0 ,r)

1/p |ui − u(x0 )| dµ . p

B(x0 ,r)

Let i = 1, 2, . . . be large enough so that x0 ∈ Wi . Then the approximate limit of u − ui at x0 is zero. Therefore Lemma 3.1 implies that (3.5) 1/p Z kD(u − ui )k(B(x0 , τ r)) p |u − ui | dµ . lim sup ≤ c lim sup r µ(B(x0 , r)) r→0 r→0 B(x0 ,r) Let

n o kD(u − ui )k(B(x, τ r)) Zi = x ∈ X : lim sup r ≤ε . µ(B(x, r)) r→0 We begin by estimating the first term on the right hand side of (3.4) in the case that x0 ∈ Z, where Z = ∪∞ i=1 Zi . Observe that Zi ⊂ Zi+1 , which follows from the coarea formula using a similar argument as in the end of this proof. Then the definition of Zi together with (3.5) shows that for large i, Z 1/p lim |u − ui |p dµ ≤ cε. r→0

B(x0 ,r)

For the second term on the right hand side of (3.4), we have Z 1/p Z 1/p p |ui − u(x0 )| dµ ≤ |ui − ui (x0 )|p dµ B(x0 ,r)

B(x0 ,r)

+ |ui (x0 ) − u(x0 )|. Since i ≥ |u(x0 )|, we have |ui (x0 ) − u(x0 )| = 0. On the other hand, Z |ui − ui (x0 )|p dµ → 0 B(x0 ,r)

as r → 0 by Lemma 3.4 because ui is bounded and u ei (x0 ) = u(x0 ).

12


Finally, we will show that H(X \ Z) = 0. By Lemma 2.6, c (3.6) H(X \ Zi ) ≤ kD(u − ui )k(X). ε For t < 0 we have u − ui > t if and only if u > t − i, and for t > 0 we have u − ui > t if and only if u > t + i. Using the coarea formula (2.5) and the fact that a complement of a set has the same perimeter as the set itself, we obtain Z ∞ P ({u − ui > t}, X) dt kD(u − ui )k(X) = −∞

Z

0

P ({u − ui ≤ t}, X) dt +

= −∞ Z 0

(3.7)

∞

Z

P ({u − ui > t}, X) dt 0

Z

∞

P ({u ≤ −i + t}, X) dt +

= −∞

P ({u > i + t}, X) dt 0

Z P ({u > t}, X) dt.

= {|t|>i}

Since u ∈ BV (X), the previous estimate implies that kD(u − ui )k(X) → 0 as i → ∞. Consequently, estimate (3.6) shows that H(X \ Z) ≤ H(X \ Zi ) → 0 as i → ∞, and the theorem follows.

Now we are ready to complete the proof of our main result. Proof of Theorem 1.1. As before, denote Et = {u > t}. Let [ P = ∂ ∗ Et \ Σγ (Et ) , t∈Q

where Σ and γ are as in Theorem 2.4. Theorem 2.4 then implies that H(P ) = 0. Fix x ∈ Su \ P . Without loss of generality, we can scale the function so that u∧ (x) = 0 and u∨ (x) = 1. We set v = (u − 1)+ − u− . Then x ∈ X \ Sv and v(x) = 0. Now take t ∈ Q such that 0 < t < 1. By the choice of P , we have Z lim inf χEt dµ ≥ γ. r→0

B(x,r)

As u ≥ v + tχEt , we have Z Z u dµ ≥ lim inf lim inf r→0

B(x,r)

r→0

Z v dµ + t lim inf B(x,r)

r→0

χEt dµ ≥ tγ. B(x,r)

Here we used Theorem 3.5 for v. The claim follows by letting t → 1. By symmetry, we also have the upper bound.


13

The next example shows that, unlike in the classical Euclidean setting, we cannot hope to get the existence of the Lebesgue points Halmost everywhere in X. However, Theorem 1.1 indicates that both the limit supremum and the limit infimum are comparable to u H-almost everywhere in X. Example 3.6. Let X = R2 be equipped with the Euclidean metric and the weighted measure dµ = ω dL2 with the weight ω given as follows. For n = 1, 2, . . . let An = {(x, y) ∈ R2 : 4−n < |y| ≤ 41−n }, and define X

ω = 1 + χ{y0}

n even

X

χ An .

n odd

Then X is a metric measure space with a doubling measure, which is even Ahlfors 2-regular, and supports a (1, 1)-Poincaré inequality. However, the characteristic function u of the set {(x, y) ∈ R2 : |x| ≤ 1, −1 ≤ y ≤ 0} is in BV (X), and for every z = (x, 0) with |x| < 1, we have Z 1 u dµ = + α lim sup 2 r→0 B(z,r) and

Z lim inf r→0

for some constant 0 < α
0. Let E0 be the collection of points x ∈ E for which lim inf r→0

µ(B(x, r) ∩ E) > 0. µ(B(x, r))

14


Note that by the Lebesgue differentiation theorem we have µ(E \E0 ) = 0 and hence kDuk(E \ E0 ) = 0. Therefore it suffices to prove that kDuk(E0 ) = 0. In oder to see this, we apply the coarea formula. Denote Et = {u > t}. Let t > 0 be such that Et has finite perimeter in X. This is possible by the coarea formula. If H(E0 ∩ ∂ ∗ Et ) > 0, then because E0 ∩ ∂ ∗ Et ⊂ Su , the measure kDuk cannot be absolutely continuous with respect to µ on E0 and hence on E, see for example [3, Theorem 5.3 and Theorem 4.4]. Therefore for all such t > 0 we have H(E0 ∩ ∂ ∗ Et ) = 0. Finally, by the coarea formula again, Z ∞ Z ∞ 0 ≤ kDuk(E0 ) = P (Et , E0 ) dt ≤ c H(E0 ∩ ∂ ∗ Et ) dt = 0. 0

0

It would also be interesting to know, more generally, if E is a Borel set on which kDuk is absolutely continuous with respect to µ, then u|E ∈ N 1,1 (E) with the weak upper gradient g = dkDuk/dµ. 4. Approximations for BV functions and the Leibniz rule As an application of Theorem 1.1, we study approximations of BV functions. By the definition of functions in BV (X), we know that given u ∈ BV (X) we can find a sequence of functions in N 1,1 (X), or a sequence of locally Lipschitz continuous functions φk , k = 1, 2, . . . , that converges to u in L1 (X) and with kDφk k converging weakly to the Radon measure kDuk. The problem with this sequence is that we do not have control over how well it converges to u pointwise. In this section we give two different constructions of approximating sequences that converge to u ∈ BV (X); unfortunately, we lose the precise control over the weak convergence of the measures. However, in the event that the goal is to study capacities of sets, the needed control is over pointwise convergence, and in this case either of these two propositions would be useful. Recall, the pointwise representative u of u is defined in (1.2). Proposition 4.1. Let u ∈ BV (X). Then there is a sequence of locally Lipschitz functions uk , k = 1, 2, . . . , such that uk → u H-almost everywhere in X \ Su , uk → u in L1 (X) as k → ∞, and lim sup kDuk k(X) ≤ c kDuk(X). k→∞

Moreover, we have 1 (1 − γ)u∧ (x) + γu∨ (x) ≤ lim inf uk (x) k→∞ c ≤ lim sup uk (x) ≤ c γu∧ (x) + (1 − γ)u∨ (x) . k→∞


15

for H-almost every x ∈ Su . Here γ with 0 < γ ≤ 21 and c > 0 depend only on the doubling constant and the constants in the in the Poincaré inequality. Proof. Let u ∈ BV (X), and ε > 0. Since u = u+ − u− , we may assume without loss of generality that u ≥ 0. Because the measure on X is doubling, we can cover X by a countable collection Bi , i = 1, 2, . . . , of balls Bi = B(xi , ε) such that ∞ X

χ20τ Bi ≤ c,

i=1

where c depends solely on the doubling constant and τ is the constant in the Poincaré inequality. Subordinate to this cover, there is a Lipschitz P∞ partition of unity ϕi with 0 ≤ ϕi ≤ 1, i=1 ϕi = 1, ϕi is C/ε-Lipschitz, and supp(ϕi ) ⊂ 2Bi for every i = 1, 2, . . . . We set ∞ X

uε =

u5Bi ϕi .

i=1

Sometimes uε is called the discrete convolution of u. For x ∈ Bj , note that ∞ X X |u(x) − u5Bi |. |uε (x) − u(x)| = |u5Bi − u(x)|ϕi (x) ≤ i=1

{i:2Bi ∩Bj 6=∅}

Hence Z |uε − u| dµ ≤ Bj

Z

X

|u − u5Bi | dµ

{i:2Bi ∩Bj 6=∅}

X

≤c

{i:2Bi ∩Bj 6=∅}

Bj

Z |u − u10Bj | dµ 10Bj

Z ≤c

|u − u10Bj | dµ 10Bj

≤ cε kDuk(10τ Bj ), where we used the bounded overlap property of the collection 20τ Bi , i = 1, 2, . . . By the bounded overlap property again and the Poincaré inequality, we see that Z ∞ Z X |uε − u| dµ ≤ |uε − u| dµ X

j=1

≤ cε

Bj

∞ X

kDuk(10τ Bj ) ≤ cε kDuk(X).

j=1

Thus uε → u in L1 (X) as ε → 0.

16


Next, if x, y ∈ Bj , then by standard arguments, and noting that the radius of 5Bj is 5ε, |uε (x) − uε (y)| ≤

∞ X

|u5Bi − u5Bj | |ϕi (x) − ϕi (y)|

i=1

c ≤ d(x, y) ε ≤

c d(x, y) ε

= c d(x, y)

X

|u5Bi − u5Bj |

{i:2Bi ∩Bj 6=∅}

Z |u − u10Bj | dµ 10Bj

kDuk(10τ Bj ) , µ(Bj )

and so uε is locally Lipschitz continuous with an upper gradient ∞ X kDuk(10τ Bj ) χBj . (4.1) gε = c µ(B ) j j=1 Hence by the bounded overlap property of the cover again and by the fact that if Bj intersects Bk then µ(Bj ) ≈ µ(Bk ), we have Z ∞ Z X gε dµ ≤ gε dµ kDuε k(X) ≤ X

≤c

k=1

∞ X

X

Bk

kDuk(10τ Bj )

k=1 {j:Bj ∩Bk 6=∅}

≤c

∞ X

kDuk(20τ Bk ) ≤ c kDuk(X).

k=1

Finally, we show estimates for the pointwise limes superior and limes inferior. To do so, we fix x ∈ X for which the conclusion of Theorem 1.1 holds and we obtain Z ∧ ∨ (1 − γ)u (x) + γu (x)) ≤ lim inf u dµ r→0 B(x,r) Z ≤ lim sup u dµ ≤ γu∧ (x) + (1 − γ)u∨ (x). r→0

B(x,r)

Furthermore, if x ∈ 2Bi (and hence ϕi (x) 6= 0), we have Z Z 1 u dµ ≤ u5Bi ≤ c u dµ. c B(x,3ε) B(x,7ε) Fix δ > 0. For sufficiently small ε, we have by the fact that 1, 1 (1 − γ)u∧ (x) + γu∨ (x) − δ ≤ uε (x) c ≤ c γu∧ (x) + (1 − γ)u∨ (x) + δ,

P∞

i=1

ϕi =


17

Since this holds for every δ > 0, it follows that 1 (1 − γ)u∧ (x) + γu∨ (x) ≤ lim inf uε (x) ε→0 c ≤ lim sup uε (x) ≤ c γu∧ (x) + (1 − γ)u∨ (x) . ε→0

As a consequence of the above approximation we have a version of Leibniz rule for nonnegative and bounded functions in BV (X). It also holds for more general bounded functions by replacing u with u+ + u− and correspondingly for v. Corollary 4.2. There is a constant c ≥ 1 such that whenever u, v ∈ BV (X) are nonnegative and bounded functions, then uv ∈ BV (X) with dkD(uv)k ≤ cu dkDvk + cv dkDuk. Here the constant c > 0 depends only on the doubling constant and the constants in the Poincaré inequality. Proof. For ε > 0 let vε be the approximation of v as in Proposition 4.1 above and let gε be an upper gradient of vε as in (4.1). Moreover, let uk , k = 1, 2, . . . , be an approximation of u in the sense that uk are nonnegative Lipschitz functions bounded by kukL∞ (X) with uk → u in L1 (X) and kDuk k converges weakly to kDuk as k → ∞. Now, it is clear that uk vε forms an approximation of uv in L1 (X). It follows that kD(uv)k ≤ lim inf lim inf kD(uk vε )k. ε→0

k→∞

1,1 Note that by the Leibniz rule for functions in Nloc (X),

dkD(uk vε )k ≤ vε dkDuk k + uk gε dµ. Let us first consider the term vε dkDuk k. Since vε is continuous on X and kDuk k converges weakly to kDuk, we have that as k → ∞ the Radon measures vε dkDuk k converges weakly to vε dkDuk. Now, as ε → 0 we have by Theorem 1.1 that lim sup vε ≤ 2c(1 − γ) v ε→0

H-almost everywhere in X and hence kDuk-almost everywhere in X. Thus whenever φ is a compactly supported Lipschitz function on X, by the dominated convergence theorem we have Z Z lim inf φvε dkDuk ≤ c φv dkDuk, ε→0

X

X

that is, a weak limit of the sequence vε dkDuk is dominated by c v dkDuk. Next we consider the term uk gε dµ as first k → ∞ and then ε → 0. Since uk → u µ-almost everywhere in X and in L1 (X), we see that

18


whenever φ is a compactly supported Lipschitz function on X, Z Z lim φuk gε dµ = φu gε dµ. k→∞

X

X

By the definition of gε considered in Proposition 4.1, using the cover Bi , i = 1, 2, . . . , from this proposition, we see that when φ is also nonnegative, Z ∞ Z X φu gε dµ ≈ φu gε dµ X

≈

i=1 Bi ∞ Z X

φu dµ kDvk(10τ Bi )

i=1 Bi ∞ Z X

≤c

φu dµ kDvk(10τ Bi ).

i=1 20τ Bi

Here we used the definition of gε (4.1). Let ψε be a function defined on X by ∞ Z X ψε (x) = φu dµ χ10τ Bi . i=1 40τ B(x,ε)

By the bounded overlap of the collection 10τ Bi , i = 1, 2, . . . , we get Z Z φugε dµ ≤ c ψε dkDvk. X

X

As φ is continuous, Theorem 1.1 shows that lim sup ψε ≤ c φu ε→0

H-almost everywhere in X and hence kDvk-almost everywhere. Hence we get Z Z lim sup lim φugε dµ ≤ c φu dkDvk k→∞

ε→0

X

X

whenever φ is a nonnegative compactly supported Lipschitz continuous function on X. It follows that if ν is a Radon measure that is obtained as a weak limit of a subsequence of the measures uk gε dµ, then Z ν(X) = sup φ dν φ∈Lipc (X), 0≤φ≤1

X

Z =

sup

lim sup lim

φ∈Lipc (X), 0≤φ≤1

ε→0

k→∞

φugε dµ X

Z ≤c

sup φ∈Lipc (X), 0≤φ≤1

φu dkDvk. X


19

One could perform the above computation by restricting to functions φ with support in a given open set, and so we can conclude that Z ν(U ) ≤ c u dkDvk U

for all open subsets U of X, and it follows that dν ≤ c u dkDvk.

Note that if u, v ∈ N 1,1 (X) ∩ L∞ (X), then we have the strongest Leibnitz rule: dkD(uv)k ≤ u dkDvk + vdkDuk. In the Euclidean setting, this strong form persists for BV functions as well, see for example [19] or the Appendix of [5]. In the metric setting the above weak version of Leibniz rule is the best possible, that is, we cannot take c = 1, as the following example shows. Example 4.3. Let X = R2 with the Euclidean distance and the measure dµ = ω dL2 , where ω = 2 − χB(0,1) . Let u = v = χB(0,1) . Thus we have u = v = 1/3 on ∂B(0, 1). Since uv = u = v and the support of the BV measure of each function is on ∂B(0, 1), we have dkD(uv)k = dkDuk > 32 dkDuk = vdkDuk + udkDvk. We next show that Lipschitz functions are dense in BV (X) in the Lusin sense. For the Euclidean case, see [6, page 252, Theorem 2] and for Sobolev functions in metric spaces, we refer to [8] and [12]. Proposition 4.4. Let u ∈ BV (X). The for each ε > 0, there exists a Lipschitz function v : X → R such that µ {x ∈ X : u(x) 6= v(x)} < ε and ku − vkL1 (X) < ε. In addition, we have kDvk(X) ≤ c kDuk(X) for some constant c > 0 depending only on the doubling constant and the constants in the Poincaré inequality. Proof. Let λ > 0 and let o n kDuk(B) Eλ = x ∈ X : ≤ λ for all balls B 3 x . µ(B) Note that X \ Eλ is an open set, and so Eλ is a Borel set. We begin by showing that there is a constant c > 0 such that c (4.2) µ(X \ Eλ ) ≤ kDuk(X \ Eλ ). λ Let Bi = B(xi , ri ) ⊂ X \ Eλ , i = 1, 2, . . . , be disjoint balls such that X \ Eλ ⊂

∞ [ i=1

5Bi

and kDuk(Bi ) > λµ(Bi )

20


for all i = 1, 2, . . . . Since µ is doubling and kDuk is a measure, we obtain µ(X \ Eλ ) ≤

∞ X i=1

∞ c3D X c3 µ(5Bi ) ≤ kDuk(Bi ) ≤ D kDuk(X \ Eλ ). λ i=1 λ

Hence (4.2) follows. Next we will show that there is a constant c > 0 that depends only on the doubling constant cD and the constants of the Poincaré inequality, such that |u(x) − u(y)| ≤ cλd(x, y)

(4.3)

for almost every x, y ∈ Eλ . Let x, y ∈ Eλ be Lebesgue points of u. Let r = d(x, y), Bx = B(x, r) and By = B(y, r). Then |u(x) − u(y)| ≤ |u(x) − uBx | + |uBx − uBy | + |u(y) − uBy |, where, by a standard chaining argument using the doubling property and the Poincaré inequality, |u(x) − uBx | ≤

∞ X

|u2−i Bx (x) − u2−(i+1) Bx |

i=0

≤ cD ≤

∞ Z X

|u − u2−i Bx | dµ −i B 2 x i=0 ∞ X kDuk(τ 2−i Bx ) cD cP r 2−i µ(2−i Bx ) i=0 ∞ X −i

≤ crλ

2

= crλ.

i=0

Similar estimate holds for |u(y) − uBy |. By the doubling property and the Poincaré inequality, we obtain Z |uBx − uBy | ≤ |u − uBx | dµ By

Z ≤c

|u − u2Bx | dµ ≤ cr B2Bx

kDuk(τ 2Bx ) ≤ crλ. µ(2Bx )

Since µ-almost every point of u is a Lebesgue point, previous estimates imply that inequality (4.3) holds. Now the first claim of this theorem follows using (4.2), (4.3) and McShane’s extension theorem for Lipschitz functions. We next provide a control of ku − vλ kL1 (X) . Note that u = vλ on Eλ . We may assume first that u is bounded on X. Hence we can assume


21

also that |vλ | ≤ kukL∞ (X) , and so Z Z |u − vλ | dµ = |u − vλ | dµ ≤ 2kukL∞ (X) µ(X \ Eλ ) X

X\Eλ

c ≤ kukL∞ (X) kDuk(X \ Eλ ). λ Note that kDuk is absolutely continuous with respect to µ on Eλ st and, by Lemma 3.7, we have kD(u − vλ )k(X) = kD(u − vλ )k(X \ Eλ ) ≤ kDuk(X \ Eλ ) + λµ(X \ Eλ ) ≤ c kDuk(X \ Eλ ). We can remove the assumption on boundedness of u by approximating u by bounded functions un = min{n, max{u, −n}} ∈ BV (X),

n = 1, 2, . . . ,

and approximate un by Lipschitz functions as above. Observe that Z Z |u − un | dµ ≤ |u| dµ → 0 X

{|u|>n}

as n → ∞. By the same argument as in (3.7), we also have kD(u − un )k(X) → 0 as n → ∞.

Remark 4.5. By considering χB for a ball B in Rn , we see that the result above is optimal. References [1] L. Ambrosio: Fine properties of sets of finite perimeter in doubling metric measure spaces. Set valued analysis 10 (2002) no. 2–3, 111–128. [2] L. Ambrosio, N. Fusco and D. Pallara: Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 2000. [3] L. Ambrosio, M. Miranda, Jr., and D. Pallara: Special functions of bounded variation in doubling metric measure spaces. Calculus of variations: topics from the mathematical heritage of E. De Giorgi (2004), 1–45. [4] L. Ambrosio and P. Tilli: Topics on analysis in metric spaces. Oxford Lecture Series in Mathematics and its Applications, 25. Oxford University Press, Oxford, 2004. [5] C. Camfield: Comparison of BV norms in weighted Euclidean spaces and metric measure spaces, PhD Thesis, University of Cincinnati (2008), 141 pp. Available at http://etd.ohiolink.edu/view.cgi?acc num=ucin1211551579. [6] L.C. Evans and R.F. Gariepy: Measure Theory and Fine Properties of Functions. CRC Press, 1992, Boca Raton-New York-London-Tokyo. [7] H. Federer: Geometric Measure Theory. Grundlehren 153, Springer–Verlag, Berlin, 1969. [8] P. Hajlasz and J. Kinnunen, H¨ older quasicontinuity of Sobolev functions on metric spaces, Rev. Mat. Iberoamericana 14 (1998), no. 3, 601–622.

22


[9] P. Hajlasz and P. Koskela: Sobolev met Poincaré. Mem. Amer. Math. Soc. 145 (2000), no. 688. [10] J. Heinonen: Lectures on Analysis on metric spaces. Universitext, SpringerVerlag, New York, 2001. [11] J. Kinnunen and V. Latvala: Lebesgue points for Sobolev functions on metric spaces. Rev. Mat. Iberoamericana 18 (2002), 685–700. [12] J. Kinnunen, R. Korte, N. Shanmugalingam, and H. Tuominen: Lebesgue points and capacities via boxing inequality in metric spaces. Indiana Univ. Math. J. 57 (2008), no. 1, 401–430. [13] N.G. Meyers and W.P. Ziemer: Integral Inequalities of Poincaré and Wirtinger type for BV Functions. Amer. J. Math 99 (1977), no. 6, 1345– 1360. [14] M. Miranda, Jr.: Functions of bounded variation on ”good” metric spaces. J. Math. Pures Appl. (9) 82 (2003), no. 8, 975–1004. [15] P. Mattila: Geometry of sets and measures in Euclidean spaces Fractals and rectifiability. Cambridge Studies in Advanced Mathematics, 44. Cambridge University Press, Cambridge, 1995. [16] T. M¨ ak¨ al¨ ainen: Adams inequality on metric measure spaces. Rev. Mat. Iberoamericana 25 (2009), no. 2, 533–558. [17] N. Shanmugalingam: Newtonian spaces: An extension of Sobolev spaces to metric measure spaces. Rev. Mat. Iberoamericana, 16 (2000), 243–279. [18] A.I. Vol’pert: Spaces BV and quasilinear equations. (Russian) Mat. Sb. (N.S.) 73 (115) (1967), 255–302. (English) Mathematics of the USSRSbornik 2, no.2. (1967), 225–267. [19] A.I. Vol’pert and S.I. Hudjaev: Analysis in classes of discontinuous functions and equations of mathematical physics. Martinus Nijhoff, Dordrecht, 1985. [20] W.P. Ziemer: Weakly differentiable functions. Graduate Texts in Mathematics 120, Springer-Verlag, 1989.

Addresses: J.K.: Department of Mathematics, P.O. Box 11100, FI-00076 Aalto University, Finland. E-mail: [email protected] R.K.: Department of Mathematics and Statistics, P.O. Box 68 (Gustaf Hällströmin katu 2b), FI-00014 University of Helsinki, Finland. E-mail: [email protected] N.S.: Department of Mathematical Sciences, P.O. Box 210025, University of Cincinnati, Cincinnati, OH 45221–0025, U.S.A. E-mail: [email protected] H.T.: Department of Mathematics and Statistics, P.O. Box 35, FI40014 University of Jyväskylä, Finland E-mail: [email protected]