POINTWISE PROPERTIES OF FUNCTIONS OF BOUNDED VARIATION …users.jyu.fi/~tuheli/paperit/pointwiseBV_sent.pdf · pointwise properties of functions of bounded variation in metric spaces

POINTWISE PROPERTIES OF FUNCTIONS OFBOUNDED VARIATION IN METRIC SPACES

JUHA KINNUNEN, RIIKKA KORTE, NAGESWARI SHANMUGALINGAMAND HELI TUOMINEN

Abstract. We study pointwise properties of functions of boundedvariation on a metric space equipped with a doubling measureand supporting a Poincare inequality. In particular, we obtaina Lebesgue type result for BV functions. We also study approx-imations by Lipschitz continuous functions and a version of theLeibniz rule. We give examples which show that our results areoptimal for BV functions in this generality.

1. Introduction

This paper studies Lebesgue points for functions of bounded vari-ation on a metric measure space (X, d, µ) equipped with a doublingmeasure and supporting a Poincare inequality. Here the metric d andthe measure µ will be fixed, and we denote the triple (X, d, µ) simplyby X. We say that x ∈ X is a Lebesgue point of a locally integrablefunction u if the limit

limr→0

∫B(x,r)

u dµ

exists. Observe that if almost every point is a Lebesgue point of everyfunction u in a certain class of functions, then

limr→0

∫B(x,r)

|u− u(x)| dµ

for almost every x ∈ X. This follows easily by considering |u − q|with q ∈ Q. The set of non-Lebesgue points of a classical Sobolevfunction is a set of measure zero with respect the Hausdorff measure ofcodimension one and this holds true also in metric spaces supportinga doubling measure and a Poincare inequality, see [11] and [12].

The situation is more delicate for BV functions. It is known that inthe Euclidean case with Lebesgue measure, aBV function has Lebesguepoints 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 researchis supported by the Academy of Finland.

1

2 KINNUNEN, KORTE, SHANMUGALINGAM AND TUOMINEN

precisely,

(1.1) limr→0

∫B(x,r)

u dy =1

2

(u∧(x) + u∨(x)

)for Hn−1-almost every x, see [6, Corollary 1 of page 216], [7, Theorem4.5.9] and [20, Theorem 5.14.4]. Here u∧(x) and u∨(x) are the lowerand upper approximate limits of u at x, respectively. The proof of thisresult lies rather deep in the theory of BV functions and it seems to bevery sensitive to the measure. Indeed, we give a simple example whichshows that the corresponding result is not true even in the Euclideancase with a weighted measure. However, we are able to show the fol-lowing metric space analogue of the result. The Hausdorff measure ofcodimension one is denoted by H. The precise definitions will be givenin Section 2.

Theorem 1.1. Assume that µ is doubling and that X supports a weak(1, 1)-Poincare inequality. If u ∈ BV (X), then for H-almost everyx ∈ X, we have

(1− γ)u∧(x) + γu∨(x) ≤ lim infr→0

∫B(x,r)

u dµ

≤ lim supr→0

∫B(x,r)

u dµ ≤ γu∧(x) + (1− γ)u∨(x),

where 0 < γ ≤ 12

and γ depends only on the doubling constant and theconstants in the in the weak (1, 1)-Poincare inequality.

We also give an example which shows that, unlike in the classicalEuclidean setting, we cannot hope to get the existence of the Lebesguepoints H-almost everywhere in X. However, the main result aboveindicates that both limes infimum and limes supremum are comparableto the pointwise representative u defined by

(1.2) u(x) = limr→0

∫B(x,r)

u dµ

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 locallyLipschitz continuous functions in L1(X) so that the integral of uppergradients converges to the total variation measure. In some applica-tions, a better control on pointwise convergence would be desirable. Weconstruct two approximation results and apply one of the approxima-tions 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 constantappears, which is not present for Sobolev functions and we show thatthis 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 aBorel 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 ∈ Xand r > 0,

µ(B(x, 2r)) ≤ cDµ(B(x, r)).

It follows from the doubling condition that

µ(B(y,R))

µ(B(x, r))≤ c

(R

r

)Qfor every r ≤ R and y ∈ B(x, r) for some Q > 1 that only dependson cD. We recall that complete metric space endowed with a doublingmeasure is proper, that is, closed and bounded sets are compact.

A nonnegative Borel function g on X is an upper gradient of anextended real valued function u on X if for all paths γ in X we have

(2.1) |u(x)− u(y)| ≤∫γ

g ds,

whenever both u(x) and u(y) are finite, and∫γg ds = ∞ otherwise.

Here x and y are the end points of γ. If g is a nonnegative measurablefunction on X and if (2.1) holds for almost every path with respect tothe 1-modulus, then g is a 1-weak upper gradient of u. The collection ofall upper gradients, together, play the role of the modulus of the weakderivative of a Sobolev function in the metric setting. We consider thefollowing norm

‖f‖N1,1(X) := ‖f‖L1(X) + infg‖g‖L1(X)

with the infimum taken over all upper gradients g of f . The Newton-Sobolev space considered in this note is the space

N1,1(X) = {u : ‖u‖N1,1(X) <∞}/∼,where the equivalence relation ∼ is given by f ∼ h if and only if

‖f − h‖N1,1(X) = 0.

Next we recall the definition and basic properties of functions ofbounded variation on metric spaces, see [14].

Definition 2.1. For u ∈ L1loc(X), we define

‖Du‖(X)

= inf{

lim infi→∞

∫X

gui dµ : ui ∈ Liploc(X), ui → u in L1loc(X)

},

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


‖DχE‖(X) < ∞. By replacing X with an open set U ⊂ X, we maydefine ‖Du‖(U) and we denote

P (E,U) = ‖DχE‖(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

‖Du‖(A) = inf{‖Du‖(U) : U ⊃ A, U ⊂ X is open

}.

Then ‖Du‖(·) is a finite Borel outer measure.

Let E be a set of finite perimeter in X. For every set A ⊂ X, wedenote

P (E,A) = ‖DχE‖(A).

We say that X supports a weak (1, 1)-Poincare inequality if thereexist 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 gof u, we have

(2.2)

∫B

|u− uB| dµ ≤ cP r

∫τB

g dµ,

where

uB =

∫B

u dµ =1

µ(B)

∫B

u dµ.

If the space supports a weak (1, 1)-Poincare inequality, then for everyu ∈ BV (X), we have

(2.3)

∫B

|u− uB| dµ ≤ cP r‖Du‖(τB)

µ(B),

where the constant cP and the dilation factor τ are the same constantsas in (2.2).

The (1, 1)-Poincare inequality implies that

(2.4)(∫

B

|u− uB|Q/(Q−1) dµ)(Q−1)/Q

≤ cr‖Du‖(τB)

µ(B),

see [9, Theorem 9.7]. Here the constant c > 0 depends only on thedoubling constant and the constants in the Poincare inequality. Weassume, without further notice, that the measure µ is doubling andthat the space supports a weak (1, 1)-Poincare inequality

The following coarea formula holds for BV functions. For a proofsee [14, Proposition 4.2]. If u ∈ L1

loc(X) and A ⊂ X is open, then

(2.5) ‖Du‖(A) =

∫ ∞−∞

P ({u > t}, A) dt.

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


The restricted spherical Hausdorff content of codimension one on Xis

HR(E) = inf{ ∞∑

i=1

µ(B(xi, ri))

ri: E ⊂

∞⋃i=1

B(xi, ri), ri ≤ R},

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

H(E) = limR→0HR(E).

A combination of [3, Theorems 4.4 and 4.6] gives the equivalence of theperimeter measure and the Hausdorff measure of codimension one forsets of finite perimeter. The measure theoretic boundary of E, denotedby ∂∗E, is the set of points x ∈ X, where both E and its complementhave positive density, i.e.

lim supr→0

µ(E ∩B(x, r))

µ(B(x, r))> 0 and lim sup

r→0

µ(B(x, r) \ E)

µ(B(x, r))> 0.

The following result is extremely useful for us. For the proof, werefer to [1] and [3].

Theorem 2.3. Assume that E is a set of finite perimeter. Then

(2.6)1

cP (E,A) ≤ H(∂∗E ∩ A) ≤ cP (E,A),

where c depends only on the doubling constant and the constants in thePoincare inequality.

Moreover, the following theorem shows that we can consider even asmaller 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 defineΣγ(E) to be the set consisting of all points x ∈ X for which

lim infr→0

min{µ(E ∩B(x, r))

µ(B(x, r)),µ(B(x, r) \ E)

µ(B(x, r))

}≥ γ.

Then there exists γ > 0, depending only on the doubling constant andthe constants in the Poincare inequality, such that

H(∂∗E \ Σγ(E)) = 0.

Let E ⊂ X be a Borel set. The upper density of E at a point x ∈ Xis

D(E, x) = lim supr→0

µ(B(x, r) ∩ E)

µ(B(x, r))and the lower density

D(E, x) = lim infr→0

µ(B(x, r) ∩ E)

µ(B(x, r)).


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 approximatelimits.

Definition 2.5. Let u : X → [−∞,∞] be a measurable function. Theupper 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(x) and calledthe approximate limit of u at x. The function u is approximatelycontinuous at x if u(x) = u(x). The approximate jump set of u is

Su = {u∧ < u∨}.

By the Lebesgue differentiation theorem, a locally integrable func-tion u is approximately continuous at µ-almost every point and henceµ(Su) = 0. A similar argument as in the classical case of [20, Re-mark 5.9.2] shows that a function u is approximately continuous atx 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 recallthe proof here to emphasize the fact that it also applies in the metriccontext.

Lemma 2.6. Let u ∈ BV (X). Let A be a Borel set such that

lim supr→0

r‖Du‖(B(x, r))

µ(B(x, r))> λ

for all x ∈ A. Then there is a constant c > 0, depending only on thedoubling constant, such that ‖Du‖(A) ≥ cλH(A).

Proof. Let ε > 0. Let U be an open set such that A ⊂ U . For eachx ∈ A there exists 0 < rx ≤ min{ε, dist(x,X \ U)}/5 such that

(2.7)µ(B(x, rx))

rx<‖Du‖(B(x, rx))

λ.

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


A ⊂ ∪∞i=15Bi. Using the doubling condition, (2.7) and the pairwisedisjointedness of the balls Bi, we obtain

Hε(A) ≤∞∑i=1

µ(5Bi)

5ri≤ c

∞∑i=1

µ(Bi)

ri

≤ c

λ

∞∑i=1

‖Du‖(Bi) ≤c

λ‖Du‖(U).

Since

‖Du‖(A) = inf{‖Du‖(U) : U is open, A ⊂ U},the claim follows by letting ε → 0 and then taking the infimum overopen sets U . �

3. Lebesgue points

In this section, we use the approximately continuous representativeu of u, and denote it by u.

Our first lemma shows that the Poincare inequality holds for smallradii without the integral average on the left-hand side if the approxi-mate 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

lim supr→0

(∫B(x0,r)

|u|Q/(Q−1) dµ)(Q−1)/Q

≤ c lim supr→0

r‖Du‖(B(x0, r))

µ(B(x0, r)).

Here the constant c > 0 depends only on the doubling constant and theconstants in the Poincare inequality.

Proof. Let ε > 0 and p = Q/(Q− 1). Since 0 is the approximate limitof 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-Poincare inequality (2.4) we con-clude that(∫

B

|u− uE|p dµ)1/p

≤ 2µ(B)

µ(E)

(∫B

|u− uB|p dµ)1/p

≤ 2

1− εcr‖Du‖(τB)

µ(B).


Hence (∫B

|u|p dµ)1/p

≤(∫

B

|u− uE|p dµ)1/p

+ |u|E

≤ 2

1− εcr‖Du‖(τB)

µ(B)+ t

for 0 < r ≤ rε. The claim follows by letting t→ 0 and then taking thelimes superior on both sides as r → 0. �

Lemma 3.2. Let u ∈ BV (X). Then −∞ < u∧(x) ≤ u∨(x) < ∞ forH-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 everypoint is a Lebesgue point of u, we have u∧(x) = u(x) for µ-almost everyx and therefore

D(Ft, x) = D(Et, x) = 1

when x is a Lebesgue point of u. The boxing inequality ([12, Remark3.3.(1)] and [16]) gives disjoint balls Bi = B(xi, τri), i = 1, 2, . . . , suchthat Ft ⊂ ∪∞i=15Bi and

∞∑i=1

µ(5Bi)

5τri≤ cP (Ft, X).

Since u has compact support, we have ri ≤ R, i = 1, 2, . . ., for R > 0large enough.

This implies that

HR(Ft) ≤ cP (Ft, X)

and

HR({u∧ =∞}) = HR

(⋂t>0

Ft

)≤ lim inf

t→∞HR(Ft)

≤ lim inft→∞

cP (Ft, X).(3.1)

As u∧ = u µ-almost everywhere, we also have

P (Et, X) = P (Ft, X)

and, by the coarea formula,∫ ∞−∞

P (Ft, X) dt =

∫ ∞−∞

P (Et, X) dt = ‖Du‖(X) <∞.

From this we conclude that

lim inft→∞

P (Ft, X) = 0.


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 showsthat

H({u∨ = −∞}) = 0.

The first part of the proof shows that u∨−u∧ is well definedH-almosteverywhere. Next we show that

H({u∨ − u∧ =∞}) = 0.

It follows from the definitions of the approximate limits and the mea-sure theoretic boundary that

At = {u∧ < t < u∨} ⊂ ∂∗Et

for every t ∈ R. Then, by the Fubini theorem,

(3.2)

∫ ∞−∞H(At) dt =

∫ ∞−∞

∫X

χAt(x) dH(x) dt

=

∫X

∫ ∞−∞

χAt(x) dt dH(x)

=

∫X

∫ u∨(x)

u∧(x)

1 dt dH(x)

=

∫X

(u∨(x)− u∧(x)) dH(x).

By (3.2), Theorem 2.3, and the coarea formula, we have∫Su

(u∨ − u∧) dH ≤∫ ∞−∞H({u∧ < t < u∨}) dt

≤∫ ∞−∞H(∂∗Et) dt

≤ c

∫ ∞−∞

P (Et, X) dt

= c‖Du‖(X) <∞.

Since u∧ = u∨ outside Su, the claim follows from this. �

The next example shows that (1.1) does not even hold for BV func-tions in weighted Euclidean spaces.

Example 3.3. Let X = R2 with the Euclidean distance and the mea-sure 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

1

2(u∧(x) + u∨(x)) =

1

26= 2

3= lim

r→0

∫B(x,r)

u dµ.


The above example also shows that we cannot always take γ = 1/2in 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 showingthat the claim of Theorem 1.1 holds outside the approximate jump setif u is bounded. Later we remove the extra assumption on boundednessby a limiting argument.

Lemma 3.4. Let u ∈ BV (X) ∩ L∞(X). Then

limr→0

∫B(x0,r)

|u− u(x0)|Q/(Q−1) dµ = 0.

for all x0 ∈ X \ Su.

Proof. Let x0 ∈ X \ Su. Since we use the representative u and u isapproximately continuous at x0, there is a measurable set A such thatx0 ∈ A, D(A, x0) = 1 and

limx→x0,x∈A

u(x) = u(x0).

Let r > 0, B = B(x0, r) and p = Q/(Q− 1). Then we have∫B

|u− u(x0)|p dµ =1

µ(B)

∫B∩A|u− u(x0)|p dµ

+1

µ(B)

∫B\A|u− u(x0)|p dµ.

(3.3)

The second term on the right hand side of (3.3) has an upper bound

1

µ(B)

∫B\A|u− u(x0)|p dµ ≤

µ(B \ A)

µ(B)2‖u‖p∞,

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 whend(x, x0) < rε and x ∈ A. Hence, we obtain

1

µ(B)

∫B∩A|u− u(x0)|p dµ ≤ εp

µ(B ∩ A)

µ(B)≤ εp

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

limr→0

∫B(x0,r)

|u− u(x0)|Q/(Q−1) dµ = 0.

for H-almost all x0 ∈ X \ Su.


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 \

∞⋃i=1

Wi

)= 0,

and hence it is enough to prove the claim for x0 ∈ (⋃∞i=1Wi) \ Su.

For i = 1, 2, . . . , let ui be a truncation of u defined as

ui(x) =

i, if u(x) > i,

u(x), if |u(x)| ≤ i,

−i, if u(x) < −i.By the Minkowski inequality, we have

(3.4)

( ∫B(x0,r)

|u− u(x0)|p dµ)1/p

≤( ∫

B(x0,r)

|u− ui|p dµ)1/p

+( ∫

B(x0,r)

|ui − u(x0)|p dµ)1/p

.

Let i = 1, 2, . . . be large enough so that x0 ∈ Wi. Then the approximatelimit of u− ui at x0 is zero. Therefore Lemma 3.1 implies that(3.5)

lim supr→0

( ∫B(x0,r)

|u− ui|p dµ)1/p

≤ c lim supr→0

r‖D(u− ui)‖(B(x0, τr))

µ(B(x0, r)).

Let

Zi ={x ∈ X : lim sup

r→0r‖D(u− ui)‖(B(x, τr))

µ(B(x, r))≤ ε}.

We begin by estimating the first term on the right hand side of (3.4)in the case that x0 ∈ Z, where Z = ∪∞i=1Zi. Observe that Zi ⊂ Zi+1,which follows from the coarea formula using a similar argument as inthe end of this proof. Then the definition of Zi together with (3.5)shows that for large i,

limr→0

(∫B(x0,r)

|u− ui|p dµ)1/p

≤ cε.

For the second term on the right hand side of (3.4), we have( ∫B(x0,r)

|ui − u(x0)|p dµ)1/p

≤( ∫

B(x0,r)

|ui − ui(x0)|p dµ)1/p

+ |ui(x0)− u(x0)|.Since i ≥ |u(x0)|, we have |ui(x0)− u(x0)| = 0. On the other hand,∫

B(x0,r)

|ui − ui(x0)|p dµ→ 0

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


Finally, we will show that H(X \ Z) = 0. By Lemma 2.6,

(3.6) H(X \ Zi) ≤c

ε‖D(u− ui)‖(X).

For t < 0 we have u− ui > t if and only if u > t− i, and for t > 0 wehave 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 theset itself, we obtain

‖D(u− ui)‖(X) =

∫ ∞−∞

P ({u− ui > t}, X) dt

=

∫ 0

−∞P ({u− ui ≤ t}, X) dt+

∫ ∞0

P ({u− ui > t}, X) dt

=

∫ 0

−∞P ({u ≤ −i+ t}, X) dt+

∫ ∞0

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

=

∫{|t|>i}

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

(3.7)

Since u ∈ BV (X), the previous estimate implies that

‖D(u− ui)‖(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 =⋃t∈Q

(∂∗Et \ Σγ(Et)

),

where Σ and γ are as in Theorem 2.4. Theorem 2.4 then implies thatH(P ) = 0.

Fix x ∈ Su \P . Without loss of generality, we can scale the functionso 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

lim infr→0

∫B(x,r)

χEt dµ ≥ γ.

As u ≥ v + tχEt , we have

lim infr→0

∫B(x,r)

u dµ ≥ lim infr→0

∫B(x,r)

v dµ+ t lim infr→0

∫B(x,r)

χEt dµ ≥ tγ.

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


The next example shows that, unlike in the classical Euclidean set-ting, we cannot hope to get the existence of the Lebesgue points H-almost everywhere in X. However, Theorem 1.1 indicates that both thelimit supremum and the limit infimum are comparable to u H-almosteverywhere in X.

Example 3.6. Let X = R2 be equipped with the Euclidean metric andthe 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

ω = 1 + χ{y<0}∑n even

χAn + χ{y>0}∑n odd

χAn .

Then X is a metric measure space with a doubling measure, whichis even Ahlfors 2-regular, and supports a (1, 1)-Poincare 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

lim supr→0

∫B(z,r)

u dµ =1

2+ α

and

lim infr→0

∫B(z,r)

u dµ =1

2− α,

for some constant 0 < α < 12.

Taking into account the pointwise behavior of BV functions, it isnatural to ask what type of behavior a BV function has on the setwhere its total variation measure is absolutely continuous with respectto the underlying measure µ. In particular, the following lemma tells usthat the total variation measure behaves like a weak derivative on thesubset where the function vanishes. In the Euclidean setting the abovelemma follows from [6, page 232, Theorem 3 and page 233, Theorem 4remark (i)]. The proof we give here is more direct.

Lemma 3.7. Let u ∈ BV (X) and E ⊂ X a Borel set such that ‖Du‖is absolutely continuous with respect to µ on E and u = 0 on E. Then‖Du‖(E) = 0.

Proof. Without loss of generality we may assume that u ≥ 0 on X.Since u = 0 on E, we see that ‖Du‖(int(E)) = 0. Consequently, if ∂Ehas µ-measure zero, the conclusion follows. Hence we may assume thatµ(∂E) > 0. Let E0 be the collection of points x ∈ E for which

lim infr→0

µ(B(x, r) ∩ E)

µ(B(x, r))> 0.


Note that by the Lebesgue differentiation theorem we have µ(E \E0) =0 and hence ‖Du‖(E \ E0) = 0. Therefore it suffices to prove that‖Du‖(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 perimeterin X. This is possible by the coarea formula. If H(E0 ∩ ∂∗Et) > 0,then because E0 ∩ ∂∗Et ⊂ Su, the measure ‖Du‖ cannot be absolutelycontinuous 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 haveH(E0 ∩ ∂∗Et) = 0. Finally, by the coarea formula again,

0 ≤ ‖Du‖(E0) =

∫ ∞0

P (Et, E0) dt ≤ c

∫ ∞0

H(E0 ∩ ∂∗Et) dt = 0.

�

It would also be interesting to know, more generally, if E is a Borelset on which ‖Du‖ is absolutely continuous with respect to µ, thenu|E ∈ N1,1(E) with the weak upper gradient g = d‖Du‖/dµ.

4. Approximations for BV functions and the Leibniz rule

As an application of Theorem 1.1, we study approximations of BVfunctions. By the definition of functions in BV (X), we know thatgiven u ∈ BV (X) we can find a sequence of functions in N1,1(X), ora sequence of locally Lipschitz continuous functions φk, k = 1, 2, . . . ,that converges to u in L1(X) and with ‖Dφk‖ converging weakly tothe Radon measure ‖Du‖. The problem with this sequence is that wedo not have control over how well it converges to u pointwise. In thissection we give two different constructions of approximating sequencesthat converge to u ∈ BV (X); unfortunately, we lose the precise controlover the weak convergence of the measures. However, in the eventthat the goal is to study capacities of sets, the needed control is overpointwise convergence, and in this case either of these two propositionswould be useful. Recall, the pointwise representative u of u is definedin (1.2).

Proposition 4.1. Let u ∈ BV (X). Then there is a sequence of lo-cally Lipschitz functions uk, k = 1, 2, . . . , such that uk → u H-almosteverywhere in X \ Su, uk → u in L1(X) as k →∞, and

lim supk→∞

‖Duk‖(X) ≤ c ‖Du‖(X).

Moreover, we have

1

c

((1− γ)u∧(x) + γu∨(x)

)≤ lim inf

k→∞uk(x)

≤ lim supk→∞

uk(x) ≤ c(γu∧(x) + (1− γ)u∨(x)

).


for H-almost every x ∈ Su. Here γ with 0 < γ ≤ 12

and c > 0 dependonly on the doubling constant and the constants in the in the Poincareinequality.

Proof. Let u ∈ BV (X), and ε > 0. Since u = u+−u−, we may assumewithout loss of generality that u ≥ 0. Because the measure on X isdoubling, we can cover X by a countable collection Bi, i = 1, 2, . . . , ofballs Bi = B(xi, ε) such that

∞∑i=1

χ20τBi ≤ c,

where c depends solely on the doubling constant and τ is the constant inthe Poincare inequality. Subordinate to this cover, there is a Lipschitzpartition 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

uε =∞∑i=1

u5Bi ϕi.

Sometimes uε is called the discrete convolution of u. For x ∈ Bj, notethat

|uε(x)− u(x)| =∞∑i=1

|u5Bi − u(x)|ϕi(x) ≤∑

{i:2Bi∩Bj 6=∅}

|u(x)− u5Bi |.

Hence ∫Bj

|uε − u| dµ ≤∑

{i:2Bi∩Bj 6=∅}

∫Bj

|u− u5Bi | dµ

≤ c∑

{i:2Bi∩Bj 6=∅}

∫10Bj

|u− u10Bj | dµ

≤ c

∫10Bj

|u− u10Bj | dµ

≤ cε ‖Du‖(10τBj),

where we used the bounded overlap property of the collection 20τBi,i = 1, 2, . . . By the bounded overlap property again and the Poincareinequality, we see that∫

X

|uε − u| dµ ≤∞∑j=1

∫Bj

|uε − u| dµ

≤ cε

∞∑j=1

‖Du‖(10τBj) ≤ cε ‖Du‖(X).

Thus uε → u in L1(X) as ε→ 0.


Next, if x, y ∈ Bj, then by standard arguments, and noting that theradius of 5Bj is 5ε,

|uε(x)− uε(y)| ≤∞∑i=1

|u5Bi − u5Bj | |ϕi(x)− ϕi(y)|

≤ c

εd(x, y)

∑{i:2Bi∩Bj 6=∅}

|u5Bi − u5Bj |

≤ c

εd(x, y)

∫10Bj

|u− u10Bj | dµ

= c d(x, y)‖Du‖(10τBj)

µ(Bj),

and so uε is locally Lipschitz continuous with an upper gradient

(4.1) gε = c∞∑j=1

‖Du‖(10τBj)

µ(Bj)χBj .

Hence by the bounded overlap property of the cover again and by thefact that if Bj intersects Bk then µ(Bj) ≈ µ(Bk), we have

‖Duε‖(X) ≤∫X

gε dµ ≤∞∑k=1

∫Bk

gε dµ

≤ c∞∑k=1

∑{j:Bj∩Bk 6=∅}

‖Du‖(10τBj)

≤ c∞∑k=1

‖Du‖(20τBk) ≤ c ‖Du‖(X).

Finally, we show estimates for the pointwise limes superior and limesinferior. To do so, we fix x ∈ X for which the conclusion of Theorem 1.1holds and we obtain

(1− γ)u∧(x) + γu∨(x)) ≤ lim infr→0

∫B(x,r)

u dµ

≤ lim supr→0

∫B(x,r)

u dµ ≤ γu∧(x) + (1− γ)u∨(x).

Furthermore, if x ∈ 2Bi (and hence ϕi(x) 6= 0), we have

1

c

∫B(x,3ε)

u dµ ≤ u5Bi ≤ c

∫B(x,7ε)

u dµ.

Fix δ > 0. For sufficiently small ε, we have by the fact that∑∞

i=1 ϕi =1,

1

c

((1− γ)u∧(x) + γu∨(x)

)− δ ≤ uε(x)

≤ c(γu∧(x) + (1− γ)u∨(x)

)+ δ,


Since this holds for every δ > 0, it follows that

1

c

((1− γ)u∧(x) + γu∨(x)

)≤ lim inf

ε→0uε(x)

≤ lim supε→0

uε(x) ≤ c(γu∧(x) + (1− γ)u∨(x)

).

�

As a consequence of the above approximation we have a version ofLeibniz rule for nonnegative and bounded functions in BV (X). It alsoholds 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

d‖D(uv)‖ ≤ cu d‖Dv‖+ cv d‖Du‖.

Here the constant c > 0 depends only on the doubling constant and theconstants in the Poincare inequality.

Proof. For ε > 0 let vε be the approximation of v as in Proposition 4.1above and let gε be an upper gradient of vε as in (4.1). Moreover, letuk, k = 1, 2, . . . , be an approximation of u in the sense that uk arenonnegative Lipschitz functions bounded by ‖u‖L∞(X) with uk → u inL1(X) and ‖Duk‖ converges weakly to ‖Du‖ as k → ∞. Now, it isclear that ukvε forms an approximation of uv in L1(X). It follows that

‖D(uv)‖ ≤ lim infε→0

lim infk→∞

‖D(ukvε)‖.

Note that by the Leibniz rule for functions in N1,1loc (X),

d‖D(ukvε)‖ ≤ vε d‖Duk‖+ ukgε dµ.

Let us first consider the term vε d‖Duk‖. Since vε is continuous onX and ‖Duk‖ converges weakly to ‖Du‖, we have that as k →∞ theRadon measures vε d‖Duk‖ converges weakly to vε d‖Du‖. Now, asε→ 0 we have by Theorem 1.1 that

lim supε→0

vε ≤ 2c(1− γ) v

H-almost everywhere in X and hence ‖Du‖-almost everywhere in X.Thus whenever φ is a compactly supported Lipschitz function on X,by the dominated convergence theorem we have

lim infε→0

∫X

φvε d‖Du‖ ≤ c

∫X

φv d‖Du‖,

that is, a weak limit of the sequence vε d‖Du‖ is dominated by c v d‖Du‖.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


whenever φ is a compactly supported Lipschitz function on X,

limk→∞

∫X

φuk gε dµ =

∫X

φu gε dµ.

By the definition of gε considered in Proposition 4.1, using the coverBi, i = 1, 2, . . . , from this proposition, we see that when φ is alsononnegative,∫

X

φu gε dµ ≈∞∑i=1

∫Bi

φu gε dµ

≈∞∑i=1

∫Bi

φu dµ ‖Dv‖(10τBi)

≤ c

∞∑i=1

∫20τBi

φu dµ ‖Dv‖(10τBi).

Here we used the definition of gε (4.1). Let ψε be a function defined onX by

ψε(x) =∞∑i=1

∫40τB(x,ε)

φu dµχ10τBi .

By the bounded overlap of the collection 10τBi, i = 1, 2, . . . , we get∫X

φugε dµ ≤ c

∫X

ψε d‖Dv‖.

As φ is continuous, Theorem 1.1 shows that

lim supε→0

ψε ≤ c φu

H-almost everywhere in X and hence ‖Dv‖-almost everywhere. Hencewe get

lim supε→0

limk→∞

∫X

φugε dµ ≤ c

∫X

φu d‖Dv‖

whenever φ is a nonnegative compactly supported Lipschitz continuousfunction on X. It follows that if ν is a Radon measure that is obtainedas a weak limit of a subsequence of the measures uk gεdµ, then

ν(X) = supφ∈Lipc(X), 0≤φ≤1

∫X

φ dν

= supφ∈Lipc(X), 0≤φ≤1

lim supε→0

limk→∞

∫X

φugε dµ

≤ c supφ∈Lipc(X), 0≤φ≤1

∫X

φu d‖Dv‖.


One could perform the above computation by restricting to functionsφ with support in a given open set, and so we can conclude that

ν(U) ≤ c

∫U

u d‖Dv‖

for all open subsets U of X, and it follows that dν ≤ c u d‖Dv‖. �

Note that if u, v ∈ N1,1(X) ∩ L∞(X), then we have the strongestLeibnitz rule:

d‖D(uv)‖ ≤ u d‖Dv‖+ vd‖Du‖.In the Euclidean setting, this strong form persists for BV functions aswell, see for example [19] or the Appendix of [5]. In the metric settingthe above weak version of Leibniz rule is the best possible, that is, wecannot take c = 1, as the following example shows.

Example 4.3. Let X = R2 with the Euclidean distance and the mea-sure dµ = ω dL2, where ω = 2− χB(0,1). Let u = v = χB(0,1). Thus wehave u = v = 1/3 on ∂B(0, 1). Since uv = u = v and the support ofthe BV measure of each function is on ∂B(0, 1), we have

d‖D(uv)‖ = d‖Du‖ > 23d‖Du‖ = vd‖Du‖+ ud‖Dv‖.

We next show that Lipschitz functions are dense in BV (X) in theLusin sense. For the Euclidean case, see [6, page 252, Theorem 2] andfor 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 aLipschitz function v : X → R such that

µ({x ∈ X : u(x) 6= v(x)}

)< ε

and ‖u−v‖L1(X) < ε. In addition, we have ‖Dv‖(X) ≤ c ‖Du‖(X) forsome constant c > 0 depending only on the doubling constant and theconstants in the Poincare inequality.

Proof. Let λ > 0 and let

Eλ ={x ∈ X :

‖Du‖(B)

µ(B)≤ λ for all balls B 3 x

}.

Note that X \ Eλ is an open set, and so Eλ is a Borel set. We beginby showing that there is a constant c > 0 such that

(4.2) µ(X \ Eλ) ≤c

λ‖Du‖(X \ Eλ).

Let Bi = B(xi, ri) ⊂ X \ Eλ, i = 1, 2, . . . , be disjoint balls such that

X \ Eλ ⊂∞⋃i=1

5Bi and ‖Du‖(Bi) > λµ(Bi)


for all i = 1, 2, . . . . Since µ is doubling and ‖Du‖ is a measure, weobtain

µ(X \ Eλ) ≤∞∑i=1

µ(5Bi) ≤c3Dλ

∞∑i=1

‖Du‖(Bi) ≤c3Dλ‖Du‖(X \ Eλ).

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 Poincare inequality,such that

(4.3) |u(x)− u(y)| ≤ cλd(x, y)

for almost every x, y ∈ Eλ. Let x, y ∈ Eλ be Lebesgue points of u. Letr = 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 propertyand the Poincare inequality,

|u(x)− uBx| ≤∞∑i=0

|u2−iBx(x)− u2−(i+1)Bx |

≤ cD

∞∑i=0

∫2−iBx

|u− u2−iBx| dµ

≤ cDcP r∞∑i=0

2−i‖Du‖(τ2−iBx)

µ(2−iBx)

≤ crλ∞∑i=0

2−i = crλ.

Similar estimate holds for |u(y)− uBy |. By the doubling property andthe Poincare inequality, we obtain

|uBx − uBy | ≤∫By

|u− uBx| dµ

≤ c

∫B2Bx

|u− u2Bx| dµ ≤ cr‖Du‖(τ2Bx)

µ(2Bx)≤ crλ.

Since µ-almost every point of u is a Lebesgue point, previous estimatesimply that inequality (4.3) holds. Now the first claim of this theoremfollows using (4.2), (4.3) and McShane’s extension theorem for Lips-chitz functions.

We next provide a control of ‖u− vλ‖L1(X). Note that u = vλ on Eλ.We may assume first that u is bounded on X. Hence we can assume


also that |vλ| ≤ ‖u‖L∞(X), and so∫X

|u− vλ| dµ =

∫X\Eλ

|u− vλ| dµ ≤ 2‖u‖L∞(X)µ(X \ Eλ)

≤ c

λ‖u‖L∞(X) ‖Du‖(X \ Eλ).

Note that ‖Du‖ is absolutely continuous with respect to µ on Eλstand, by Lemma 3.7, we have

‖D(u− vλ)‖(X) = ‖D(u− vλ)‖(X \ Eλ)≤ ‖Du‖(X \ Eλ) + λµ(X \ Eλ)≤ c ‖Du‖(X \ Eλ).

We can remove the assumption on boundedness of u by approximat-ing 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∫X

|u− un| dµ ≤∫{|u|>n}

|u| dµ→ 0

as n→∞. By the same argument as in (3.7), we also have

‖D(u− un)‖(X)→ 0

as n→∞. �

Remark 4.5. By considering χB for a ball B in Rn, we see that theresult above is optimal.

References

[1] L. Ambrosio: Fine properties of sets of finite perimeter in doubling metricmeasure spaces. Set valued analysis 10 (2002) no. 2–3, 111–128.

[2] L. Ambrosio, N. Fusco and D. Pallara: Functions of bounded variation andfree discontinuity problems. Oxford Mathematical Monographs. The Claren-don Press, Oxford University Press, New York, 2000.

[3] L. Ambrosio, M. Miranda, Jr., and D. Pallara: Special functions of boundedvariation in doubling metric measure spaces. Calculus of variations: topicsfrom the mathematical heritage of E. De Giorgi (2004), 1–45.

[4] L. Ambrosio and P. Tilli: Topics on analysis in metric spaces. OxfordLecture Series in Mathematics and its Applications, 25. Oxford UniversityPress, Oxford, 2004.

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[10] J. Heinonen: Lectures on Analysis on metric spaces. Universitext, Springer-Verlag, New York, 2001.

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Addresses:J.K.: Department of Mathematics, P.O. Box 11100, FI-00076 AaltoUniversity, Finland.E-mail: [email protected]

R.K.: Department of Mathematics and Statistics, P.O. Box 68 (GustafHallstromin katu 2b), FI-00014 University of Helsinki, Finland.E-mail: [email protected]

N.S.: Department of Mathematical Sciences, P.O. Box 210025, Univer-sity of Cincinnati, Cincinnati, OH 45221–0025, U.S.A.E-mail: [email protected]

H.T.: Department of Mathematics and Statistics, P.O. Box 35, FI-40014 University of Jyvaskyla, FinlandE-mail: [email protected]

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