A note on the weak* and pointwise convergence of BV functions
aa r X i v : . [ m a t h . F A ] S e p A NOTE ON THE WEAK* AND POINTWISE CONVERGENCEOF BV FUNCTIONS
LISA BECK AND PANU LAHTI
Abstract.
We study pointwise convergence properties of weakly* convergingsequences { u i } i ∈ N in BV( R n ). We show that, after passage to a suitable sub-sequence (not relabeled), we have pointwise convergence u ∗ i ( x ) → u ∗ ( x ) of theprecise representatives for all x ∈ R n \ E , where the exceptional set E ⊂ R n hason the one hand Hausdorff dimension at most n −
1, and is on the other handalso negligible with respect to the Cantor part of | Du | . Furthermore, we discussthe optimality of these results. Introduction
Let
N, n ∈ N and consider a sequence { u i } i ∈ N of functions in [BV( R n )] N . We areinterested in studying pointwise convergence properties under different assumptionsof convergence on the sequence. In this regard, let us recall that for every function u ∈ BV( R n ) its Lebesgue representative e u is well-defined outside of the approximatediscontinuity set S u (which is of Hausdorff dimension at most n − u ∗ which provides a well-definedextension of e u to the jump set J u ⊂ S u (and the remaining set S u \ J u is negligiblewith respect to the ( n − H n − ), see Section 2 forthe precise statement. If we assume the strong convergence u i → u in [ L ( R n )] N ,then it is well-known that for a (not relabeled) subsequence we have e u i ( x ) → e u ( x ) for L n -almost every x ∈ R n . If we even have strong convergence u i → u in [BV( R n )] N ,then for a (not relabeled) subsequence we have u ∗ i ( x ) → u ∗ ( x ) for H n − -almost every x ∈ R n (which follows e.g. from [11, Remark 4.1, Lemma 4.2]).Here we investigate what can be said about pointwise convergence if the sequence { u i } i ∈ N is known to converge to u in a stronger topology than [ L ( R n )] N , but aweaker one than [BV( R n )] N . Mostly, we are interested in the case of weak* conver-gence u i → u in [BV( R n )] N . For this purpose, we proceed in two different directions.First, we follow an approach via capacity estimates and prove for a subsequence thatpointwise convergence holds outside of an exceptional set E ⊂ R n of Hausdorff di-mension at most n − Theorem 1.1.
Let u ∈ BV( R n ) . Let { u i } i ∈ N be a sequence in BV( R n ) for which {| Du i | ( R n ) } i ∈ N is bounded, and suppose that e u i ( x ) → e u ( x ) for L n -almost every x ∈ R n . Then there exists a set E ⊂ R n such that dim H ( E ) ≤ n − and such thatfor a subsequence ( not relabeled ) we have e u i ( x ) → e u ( x ) for every x ∈ R n \ E. Mathematics Subject Classification.
Primary 26B30, 28A20, 31C40; Secondary 28A78.
Key words and phrases. functions of bounded variation, weak star convergence, pointwise con-vergence, variation measure, Cantor part, Hausdorff dimension.
This result is analogous to the case of weakly convergent sequences in a Sobolevspace W ,p ( R n ) with p ≥
1, where pointwise convergence of a subsequence holdsoutside of an exceptional set of Hausdorff dimension at most n − p . The proof ofTheorem 1.1 is carried out in Section 3 (along with the proof of the correspondingresult for weakly convergent sequences in Sobolev spaces as mentioned above, whichis included for comparison) and it is essentially based on capacity estimates forfractional Sobolev spaces and interpolation arguments.Second, we address pointwise convergence with respect to a diffuse measure | D d w | ,which is defined as the sum of the absolutely continuous and the Cantor part of thevariation measure | Dw | for a function w ∈ BV( R n ). We prove for a subsequence(that might depend on the choice of w ) that pointwise convergence holds outside ofan exceptional set E ⊂ R n of vanishing | D d w | -measure. Theorem 1.2.
Let u ∈ BV( R n ) . Let { u i } i ∈ N be a sequence in BV( R n ) for which {| Du i | ( R n ) } i ∈ N is bounded, and suppose that u ∗ i ( x ) → u ∗ ( x ) for L n -almost every x ∈ R n . Let w ∈ BV( R n ) . Then for a subsequence ( not relabeled ) we have u ∗ i ( x ) → u ∗ ( x ) for | D d w | -almost every x ∈ R n . We notice that the pointwise convergence with respect to the absolutely continu-ous part follows of course already from the L n -almost everywhere convergence, butthe pointwise convergence with respect to the Cantor part | D c w | may contain ad-ditional information when compared to Theorem 1.1 since | D c w | can be supportedon an ( n − Remark 1.3.
The results of Theorem 1.1 and Theorem 1.2 hold also in the vector-valued case where the sequence { u i } i ∈ N and the limit function u are taken in thespace [BV( R n )] N with N ∈ N . This is seen easily by considering the componentfunctions.Let us finally observe that the results of Theorem 1.1 and Theorem 1.2 are sharpin the sense that the exceptional set E , where pointwise convergence fails, in generaldoes not satisfy H n − ( E ) = 0 or | Dw | ( E ) = 0. In particular, we give examples inSection 5 which demonstrate that the jump set D j w needs to be excluded in thestatement of the pointwise convergence and that also the passage to a subsequenceis in general necessary. We further discuss some aspects of the possible size of theexceptional set E , and we analyze two particular situations in Section 6.2. Notation and preliminaries
General notation.
As already mentioned in the introduction, we consider
N, n ∈ N and we will always work in the space R n . The matrix space R N × n willalways be equipped with the Euclidean norm | A | := ( P Ni =1 P nj =1 A ij ) / , where i and j are the row and column indices, respectively. For a ∈ R N and b ∈ R n , wedefine the tensor product a ⊗ b := ab T ∈ R N × n , where a, b are considered as columnvectors. We write B ( x, r ) for the open ball in R n with center x and radius r , thatis, { z ∈ R n : | z − x | < r } , and we write S n − for the unit sphere in R n , that is, { z ∈ R n : | z | = 1 } . For a set S ⊂ R n we use the notation S o to indicate thetopological interior. NOTE ON THE WEAK* AND POINTWISE CONVERGENCE OF BV FUNCTIONS 3
We denote the n -dimensional Lebesgue measure by L n and use the abbreviation ω n = L n ( B (0 , s -dimensional Hausdorff measure by H s . Givenany measure ν on R n , the restriction of ν to a set S ⊂ R n is denoted by ν S , thatis, ν S ( B ) := ν ( S ∩ B ) for all sets B ⊂ R n . The Borel σ -algebra on a set S ⊂ R n isdenoted by B ( S ).For a function u we write u + := max { u, } for its positive part, and if it isintegrable on some measurable set S ⊂ R n of positive and finite Lebesgue measure,we write R S u d L n := ( L n ( S )) − R S u ( x ) d L n ( x ) for its mean value on S . We furtherdenote by S the characteristic function of the set S .2.2. Fractional Sobolev spaces and capacity.
Let Ω ⊂ R n be an open set, s ∈ (0 ,
1) and p ∈ [1 , ∞ ). A function u ∈ L p (Ω) is said to belong to the fractionalSobolev space W s,p (Ω) if ( x, z )
7→ | u ( x ) − u ( z ) || x − z | − n/p − s ∈ L p (Ω × Ω). When W s,p (Ω) is endowed with the norm k u k W s,p (Ω) := (cid:16) k u k pL p (Ω) + [ u ] pW s,p (Ω) (cid:17) p , where [ u ] pW s,p (Ω) := Z Ω Z Ω | u ( x ) − u ( z ) | p | x − z | n + sp d L n ( x ) d L n ( z ) , it is a Banach space. The number s can be interpreted as fractional differentiabil-ity, and in some sense the fractional Sobolev spaces W s,p are interpolation spacesbetween the classical Sobolev space W ,p and the Lebesgue space L p . As a matterof fact, many properties known from classical Sobolev spaces extend to fractionalSobolev spaces. We here comment only on a few on them, which are relevant forour paper, and refer for instance to [1, 6] for a detailed discussion. In particular,we have the inclusion W s ′ ,p (Ω) ⊂ W s,p (Ω) for every s ′ ∈ ( s, s ′ = 1 if Ω = R n of if Ω is a boundedLipschitz domain, see e.g. [6, Proposition 2.1 & Proposition 2.2]. Moreover, as inthe case of classical Sobolev spaces with integer differentiability and still in the caseof bounded Lipschitz domains, there exists a linear, bounded extension operatorfrom W s,p (Ω) to W s,p ( R n ), see [6, Theorem 5.4]. Furthermore, the space C ∞ c ( R n )of smooth functions with compact support is dense in W s,p ( R n ), see [1, Theorem7.38].Associated to the classical ( s = 1) and fractional ( s ∈ (0 , s, p ) -Sobolev capacity of a set E ⊂ R n ascap s,p ( E ) := inf (cid:8) k u k W s,p ( R n ) : u ∈ W s,p ( R n ) with E ⊂ { u ≥ } o (cid:9) (which as usual is interpreted as ∞ if there doesn’t exist any function u ∈ W s,p ( R n )with E ⊂ { u ≥ } o ). Note that it is easy to verify from its definition that cap s,p is anouter measure on R n , i.e., it assigns zero measure to the empty set, it is monotone,and it is countably subadditive. Moreover, there holds L n ( E ) ≤ cap s,p ( E ) for allsets E ⊂ R n , i.e., the Sobolev capacity measure cap s,p is a finer measure comparedto the Lebesgue measure L n . It is also not difficult to verify that cap s,p ( { x } ) > x ∈ R n whenever sp > n , hence, there are no nontrivialsets of vanishing cap s,p -measure, which implies that the ( s, p )-capacity is only usefulif sp ≤ n . Let us further notice that for sets of vanishing cap s,p -measure we alsohave an immediate upper bound on their Hausdorff dimension: LISA BECK AND PANU LAHTI
Proposition 2.1.
Let s ∈ (0 , and p ∈ (1 , ∞ ) with sp ≤ n . If a set E ⊂ R n satisfies cap s,p ( E ) = 0 , then we have dim H ( E ) ≤ n − sp .Proof. We essentially follow the first part of the proof of [7, Section 4.7, Theorem 4]).Since by assumption cap s,p ( E ) = 0 holds, we find a sequence of functions { u i } i ∈ N in W s,p ( R n ) such that k u i k W s,p ( R n ) ≤ − i and E ⊂ { u i ≥ } o are satisfied for each i ∈ N . The first condition ensures that v := P i ∈ N u i defines afunction in W s,p ( R n ), while the second condition implies E ⊂ (cid:26) x ∈ R n : lim sup r →∞ Z B ( x,r ) v ( z ) d L n ( z ) = ∞ (cid:27) . Since in view of [3, Proposition 1.76] the set on the right-hand side has Hausdorffdimension at most n − sp , the claim dim H ( E ) ≤ n − sp is established. (cid:3) Radon measures.
Let Ω ⊂ R n be an open set and ℓ ∈ N . We denote by C c (Ω; R ℓ ) the space of continuous R ℓ -valued functions with compact support in Ωand by C (Ω; R ℓ ) its completion with respect to the k · k ∞ -norm. We further denoteby M (Ω; R ℓ ) the Banach space of vector-valued Radon measures, equipped withthe total variation norm | µ | (Ω) < ∞ , which is defined relative to the Euclideannorm on R ℓ . By the Riesz representation theorem, M (Ω; R ℓ ) is the dual space of C (Ω; R ℓ ), with the duality pairing h φ, µ i := R Ω φ · dµ := P ℓj =1 R Ω φ j dµ j . Thus weak*convergence µ i ∗ ⇁ µ in M (Ω; R ℓ ) means h φ, µ i i → h φ, µ i for all φ ∈ C (Ω; R ℓ ). Wefurther denote the set of positive measures by M + (Ω).For a vector-valued Radon measure γ ∈ M (Ω; R ℓ ) and a positive Radon measure µ ∈ M + (Ω), we can write the Lebesgue–Radon–Nikodym decomposition γ = γ a + γ s = dγdµ dµ + γ s of γ with respect to µ , where dγdµ ∈ L (Ω , µ ; R ℓ ).For open sets E ⊂ R n − m , F ⊂ R m and m ∈ { , . . . , n − } , a parametrized measure ( ν y ) y ∈ E is a mapping from E to the set M ( F ; R ℓ ) of vector-valued Radon measureson F . It is said to be weakly* µ -measurable , for µ ∈ M + ( E ), if y ν y ( B ) is µ -measurable for all Borel sets B ∈ B ( F ) (it suffices to check this for open subsets).Equivalently, ( ν y ) y ∈ E is weakly* µ -measurable if the function y R F f ( y, t ) dν x ( t )is µ -measurable for every bounded B µ ( E ) ×B ( F )-measurable function f : E × F → R (see [2, Proposition 2.26]), where B µ ( E ) denotes the µ -completion of B ( E ). Supposethat we additionally have Z E | ν y | ( F ) dµ ( y ) < ∞ . In that case we denote by µ ⊗ ν y the generalized product measure defined by µ ⊗ ν y ( A ) := Z E (cid:18)Z F A ( y, t ) dν y ( t ) (cid:19) dµ ( y )for any A ∈ B ( E × F ). Then the integration formula Z E × F f ( y, t ) d ( µ ⊗ ν y )( y, t ) = Z E (cid:18)Z F f ( y, t ) dν y ( t ) (cid:19) dµ ( y ) NOTE ON THE WEAK* AND POINTWISE CONVERGENCE OF BV FUNCTIONS 5 holds for every bounded Borel function f : E × F → R and, in the case ℓ = 1 and ν y ≥
0, if f is a positive Borel function.2.4. Functions of bounded variation.
Let Ω ⊂ R n be an open set. A function u ∈ [ L (Ω)] N is said to belong to the space [BV(Ω)] N of functions of boundedvariation if its distributional derivative is a finite R N × n -valued Radon measure.This means that there exists a (unique) measure Du ∈ M ( R n ; R N × n ) such that forall functions ψ ∈ C c (Ω), the integration-by-parts formula Z Ω ∂ψ∂x k u j d L n = − Z Ω ψ dDu jk , j = 1 . . . N, k = 1 , . . . , n holds. The space [BV(Ω)] N is a Banach space endowed with the norm k u k [BV(Ω)] N := k u k [ L (Ω)] N + | Du | (Ω) . The theory of BV functions presented here can be found in [2] (see also [7, 8, 15]),and we will give specific references only for a few key results relevant for our paper.Let us first note that for a scalar-valued function u ∈ BV(Ω) the total variation of Du can be obtained by integration over the super-level sets S t := { x ∈ Ω : u ( x ) > t } , t ∈ R , via the coarea formula (see [2, Theorem 3.40]) as(2.1) | Du | (Ω) = Z ∞−∞ | D S t | (Ω) dt. We now recall different notions of convergence for sequences in [BV(Ω)] N . Wesay that a sequence of functions { u i } i ∈ N in [BV(Ω)] N converges weakly* to u ∈ [BV(Ω)] N , denoted by u i ∗ ⇁ u in [BV(Ω)] N , if u i → u strongly in [ L (Ω)] N and Du j ∗ ⇁ Du in M (Ω , R N × n ). Note that every weakly* converging sequence { u i } i ∈ N in [BV(Ω)] N is norm-bounded by the Banach–Steinhaus theorem. Conversely, everynorm-bounded sequence { u i } i ∈ N in [BV(Ω)] N with strong convergence u i → u in[ L (Ω)] N satisfies u i ∗ ⇁ u in [BV(Ω)] N , see [2, Proposition 3.13]. Moreover, everynorm-bounded sequence in [BV(Ω)] N has a weakly* converging subsequence if Ωis sufficiently regular, e.g. a bounded Lipschitz domain. Moreover, we say that asequence of functions { u i } i ∈ N in [BV(Ω)] N converges strictly to u ∈ [BV(Ω)] N if u i → u strongly in [ L (Ω)] N and | Du i | (Ω) → | Du | (Ω).We have the following simple fact concerning weak* convergence in the BV class. Proposition 2.2.
Let u ∈ BV( R n ) . Let { u i } i ∈ N be a sequence in BV( R n ) for which {| Du i | ( R n ) } i ∈ N is bounded, and suppose that u i ( x ) → u ( x ) for L n -almost every x ∈ R n . Then we have u i → u in L q loc ( R n ) for every q ∈ [1 , nn − ) .Proof. Fix an arbitrary
R >
0. For sufficiently large i ∈ N , we have | u i − u | ≤ A ⊂ B (0 , R ) with L n ( A ) L n ( B (0 , R )) ≥ . Therefore, by a Poincar´e inequality (see e.g. [10, Lemma 2.2]) there holds Z B (0 ,R ) | u i − u | d L n ≤ Z B (0 ,R ) ( | u i − u | − + d L n + L n ( B (0 , R )) ≤ C ( n ) R | D ( u i − u ) | ( B (0 , R )) + L n ( B (0 , R )) ≤ C ( n ) R ( | Du i | ( R n ) + | Du | ( R n )) + L n ( B (0 , R )) . LISA BECK AND PANU LAHTI
Thus, { u i − u } i ∈ N is a bounded sequence in BV( B (0 , R )). Let q ∈ [1 , nn − ) bearbitrary. By the Sobolev embedding and the Rellich–Kondrachov compactnesstheorem, we get for the sequence { u i } i ∈ N boundedness in L n/ ( n − ( B (0 , R )) and,for a subsequence, strong convergence in L q ( B (0 , R )), necessarily to 0. Since fromevery subsequence of { u i } i ∈ N we can choose a further subsequence converging to u in L q ( B (0 , R )), this is in fact true also for the original sequence. Since R > u i → u in L q loc ( R n ) as claimed. (cid:3) Let u ∈ [ L ( R n )] N . We say that u has a Lebesgue point at x ∈ R n iflim r → Z B ( x,r ) | u ( z ) − e u ( x ) | d L n ( z ) = 0for some (unique) e u ( x ) ∈ R N . We denote by S u the set where this condition fails andcall it the approximate discontinuity set . We note that S u is a Borel set of vanishing L n -measure and that e u : R n \ S u → R N is Borel-measurable, see [2, Proposition 3.64].Given ν ∈ S n − , we denote the upper and lower half-ball with respect to ν as B + ν ( x, r ) := { z ∈ B ( x, r ) : h z − x, ν i > } ,B − ν ( x, r ) := { z ∈ B ( x, r ) : h z − x, ν i < } . We say that x ∈ R n is an approximate jump point of u if there exist ν ∈ S n − and u + ( x ) , u − ( x ) ∈ R N with u + ( x ) = u − ( x ) (called the one-sided approximate limits)such that lim r → Z B + ν ( x,r ) | u ( z ) − u + ( x ) | d L n ( z ) = 0and lim r → Z B − ν ( x,r ) | u ( z ) − u − ( x ) | d L n ( z ) = 0 . We denote by J u the set of approximate jump points and call it the approximate jumpset . If u ∈ [BV(Ω)] N , then S u is countably H n − -rectifiable and H n − ( S u \ J u ) = 0,see [2, Theorem 3.78].For a finer analysis of BV functions, we can now define the precise representative of u ∈ [BV(Ω)] N as u ∗ ( x ) := (e u ( x ) if x ∈ R n \ S u , ( u + ( x ) + u − ( x )) / x ∈ J u , which is uniquely determined H n − -almost everywhere. We then write the Radon–Nikodym decomposition of the variation measure of u into the absolutely continuousand singular parts as Du = D a u + D s u . Furthermore, we define the jump and Cantorparts of Du by D j u := D s u J u , D c u := D s u ( R n \ S u ) . Since Du vanishes on H n − -negligible sets (see [2, Lemma 3.76]), we obtain with H n − ( S u \ J u ) = 0 the decomposition Du = D a u + D c u + D j u. Moreover, we call the sum D a u + D c u the diffuse part of the variation measure anddenote it by D d u . NOTE ON THE WEAK* AND POINTWISE CONVERGENCE OF BV FUNCTIONS 7
One-dimensional sections of BV functions. The following notation andresults on one-dimensional sections of BV functions, as given in [2, Section 3.11],will be crucial for us.In the one-dimensional case n = 1, we have J u = S u , J u is at most countable, and Du ( { x } ) = 0 for every x ∈ R \ J u . Moreover, we have for every x, ˜ x ∈ R \ J u with x < ˜ x that(2.2) u ∗ (˜ x ) − u ∗ ( x ) = Du (( x, ˜ x )) , and for every x, ˜ x ∈ R with x < ˜ x that(2.3) | u ∗ (˜ x ) − u ∗ ( x ) | ≤ | Du | ([ x, ˜ x ]) . In R n , we denote the standard basis vectors by e k , k = 1 , . . . , n . For any fixed k ∈ { , . . . , n } , y ∈ R n − and t ∈ R , we introduce the notation π k ( y, t ) := ( y , . . . , y k − , t, y k , . . . , y n − ) ∈ R n . For a set A ⊂ R n we then denote the slices of A at π k ( y,
0) in e k -direction by A e k y := { t ∈ R : π k ( y, t ) ∈ A } . For u ∈ [BV( R n )] N , we denote u ky ( t ) := u ( π k ( y, t )) and record that u ky ∈ [BV( R )] N is satisfied for L n − -almost every y ∈ R n − (see [2, Theorem 3.103]). Denoting D k u := h Du, e k i and D dk u := h D d u, e k i (the inner product taken row-wise), wefurther have D k u = L n − ⊗ Du ky and D dk u = L n − ⊗ D d u ky (see [2, Theorem 3.107 & Theorem 3.108]. It follows that(2.4) | D k u | = L n − ⊗ | Du ky | and | D dk u | = L n − ⊗ | D d u ky | (see [2, Corollary 2.29]). Moreover, for L n − -almost every y ∈ R n − we have(2.5) J u ky = ( J u ) ky and ( u ∗ ) ky ( t ) = ( u ky ) ∗ ( t ) for every t ∈ R \ J u ky , (see [2, Theorem 3.108]).3. Pointwise convergence w.r.t. Hausdorff measures
In this section, we first consider strongly convergent sequences in fractional Sobolevspaces W s,p ( R n ), with s ∈ (0 ,
1) and p ∈ (1 , ∞ ). For these we establish pointwiseconvergence outside of a set of vanishing ( s, p )-capacity, by a straightforward adapta-tion of the proof for classical Sobolev spaces. We then move on to bounded sequencesin the classical Sobolev spaces W ,p ( R n ) and in the space BV( R n ) of functions ofbounded variation and deduce, via compactness and interpolation results, pointwiseconvergence up to sets of Hausdorff dimension n − p and n −
1, respectively, whichyields in particular the statement of Theorem 1.1.Let us start by recalling that the exceptional set of non-Lebesgue points of a func-tion in W s,p ( R n ) is of vanishing ( s, p )-capacity (see e.g. [12, Theorem 6.2], combinedwith the argument from [7, Section 1.7, Corollary 1]), which is the analogous prop-erty as known for classical Sobolev functions (see e.g. [9] or [7, Section 4.8, Theorem1]). LISA BECK AND PANU LAHTI
Theorem 3.1.
Let s ∈ (0 , , p ∈ (1 , ∞ ) and u ∈ W s,p ( R n ) . Then there exists aset E ⊂ R n such that cap s,p ( E ) = 0 and such that for each x ∈ R n \ E , for some e u ( x ) ∈ R there holds lim r → Z B ( x,r ) | u ( z ) − e u ( x ) | d L n ( z ) = 0 . Next, following the strategy of proof in [13, Lemma 2.19], we obtain for sufficientlyfast converging sequences in W s,p ( R n ) that also the exceptional set where pointwiseconvergence fails is of vanishing ( s, p )-capacity: Lemma 3.2.
Let s ∈ (0 , , p ∈ (1 , ∞ ) and u ∈ W s,p ( R n ) . Let { u i } i ∈ N be a sequenceof functions in W s,p ( R n ) and suppose that u i → u strongly in W s,p ( R n ) with (3.1) X i ∈ N ip k u − u i k pW s,p ( R n ) < ∞ . Then there exists a set E ⊂ R n such that cap s,p ( E ) = 0 and such that for each x ∈ R n \ E the pointwise limit lim i →∞ e u i ( x ) of the Lebesgue representatives existsand coincides with e u ( x ) .Proof. In view of Theorem 3.1, we find a set E ⊂ R n with cap s,p ( E ) = 0 such thatthere hold e u i ( x ) = lim r → Z B ( x,r ) u i ( z ) d L n ( z ) and e u ( x ) = lim r → Z B ( x,r ) u ( z ) d L n ( z )for all x ∈ R n \ E and all i ∈ N . For these points x ∈ R n \ E we then observe | e u ( x ) − e u i ( x ) | ≤ sup r> Z B ( x,r ) | u ( z ) − u i ( z ) | d L n ( z ) = M ( u − u i )( x ) , where M denotes the Hardy–Littlewood maximal operator. Defining sets A i := (cid:8) x ∈ R n : M ( u − u i )( x ) > − i (cid:9) we then deduce from [12, Lemma 6.4] (based essentially on the facts that A i isan open set and that the Hardy–Littlewood maximal operator is bounded from W s,p ( R n ) into itself) the estimatecap s,p ( A i ) ≤ C ( n, p )2 ip k u − u i k pW s,p ( R n ) . Setting E := E ∪ \ j ∈ N [ i ≥ j A i , we can then verify the assertions of the lemma. First, by the choice of E and byassumption (3.1) we havecap s,p ( E ) ≤ cap s,p ( E ) + lim j →∞ X i ≥ j cap s,p ( A i ) ≤ C ( n, p ) lim j →∞ X i ≥ j ip k u − u i k pW s,p ( R n ) = 0 . Secondly, if x ∈ R n \ E , then we have x / ∈ E and there exists some j ∈ N such that x / ∈ A i for all i ≥ j . Consequently, we have M ( u − u i )( x ) ≤ − i ⇒ | e u ( x ) − e u i ( x ) | ≤ − i for all i ≥ j and thus the pointwise convergence lim i →∞ e u i ( x ) = e u ( x ) holds. This finishes theproof of the lemma. (cid:3) As a direct consequence, by passing to a sufficiently fast convergent subsequence,we have the following
Corollary 3.3.
Let s ∈ (0 , , p ∈ (1 , ∞ ) and u ∈ W s,p ( R n ) . Let { u i } i ∈ N be asequence of functions in W s,p ( R n ) and suppose that u i → u strongly in W s,p ( R n ) .Then for a subsequence ( not relabeled ) we have pointwise convergence e u i ( x ) → e u ( x ) for cap s,p -almost every x ∈ R n . With Corollary 3.3 at hand, we can now address the announced pointwise con-vergence of bounded sequences in classical Sobolev spaces W ,p ( R n ), given for thepurpose of comparison, or in the space BV( R n ) of functions of bounded variation,as stated in Theorem 1.1. Theorem 3.4.
Let p ∈ (1 , n ] and u ∈ W ,p ( R n ) . Let { u i } i ∈ N be a sequence in W ,p ( R n ) for which {k∇ u i k L p ( R n ) } i ∈ N is bounded, and suppose that e u i ( x ) → e u ( x ) for L n -almost every x ∈ R n . Then there exists a set E ⊂ R n with dim H ( E ) ≤ n − p and such that for a subsequence ( not relabeled ) we have e u i ( x ) → e u ( x ) for every x ∈ R n \ E. Before proving Theorem 3.4, let us first notice that, similarly as Proposition 2.2,we can prove the following
Proposition 3.5.
Let p ∈ (1 , n ] and u ∈ W ,p ( R n ) . Let { u i } i ∈ N be a sequence in W ,p ( R n ) for which {k∇ u i k L p ( R n ) } i ∈ N is bounded, and suppose that u i ( x ) → u ( x ) for L n -almost every x ∈ R n . Then we have u i → u in L p loc ( R n ) .Proof of Theorem 3.4. Without loss of generality we can assume that { u i } i ∈ N is asequence in W ,p ( R n ) where u i vanishes outside of some ball B (0 , R ) for all i ∈ N (otherwise, we multiply by a suitable cut-off function η R with B (0 ,R/ ≤ η R ≤ B (0 ,R ) for R ∈ N , to obtain sequences which still have norm-bounded gradients andwhich coincide on B (0 , R/
2) with the original sequence).We first observe that by Proposition 3.5, we actually have strong convergence ofthe sequence { u i } i ∈ N in L p ( R n ). We then note that for every function v ∈ W ,p ( R n )and every s ∈ (0 ,
1) the interpolation inequality[ v ] pW s,p ( R n ) ≤ p (1 − s ) nω n s (1 − s ) p k v k (1 − s ) pL p ( R n ) k∇ v k spL p ( R n ; R n ) is available, see [5, Corollary 4.2]. Therefore, we even have strong convergence ofthe sequence { u i } i ∈ N in W s,p ( R n ), for every fixed s ∈ (0 , e u i ( x ) → e u ( x ) holds outside of a set E ℓ ⊂ R n withcap − /ℓ,p ( E ℓ ) = 0, for each ℓ ∈ N . Defining the exceptional set as E := T ℓ ∈ N E ℓ , wethen have pointwise convergence e u i ( x ) → e u ( x ) for every x ∈ R n \ E . Furthermore,we deduce from Proposition 2.1 that H d ( E ) ≤ H d ( E ℓ ) = 0, for every d > n − p and ℓ > p/ ( d − n + p ). This shows that the Hausdorff dimension of E is at most n − p ,which completes the proof of the theorem. (cid:3) Remark 3.6.
Let us note that the statements of Theorem 3.1, Corollary 3.3 andTheorem 3.4 are trivially true in the case s ∈ (0 ,
1] and p ∈ [1 , ∞ ) with sp > n .By Morrey’s embedding into H¨older spaces (see [6, Theorem 8.2] for a version withfractional Sobolev spaces), the Lebesgue representative of a W s,p -function is alreadyglobally H¨older continuous. Thus, every point of a W s,p -function is a Lebesgue point,and for every bounded sequence in W s,p ( R n ) pointwise convergence of the Lebesguerepresentative of a subsequence is true on the full space, as a consequence of theArzel`a–Ascoli theorem. Proof of Theorem 1.1.
Similarly as in the proof of Theorem 3.4, we may here assumethat { u i } i ∈ N is a sequence in BV( R n ) where u i vanishes outside of some ball B (0 , R )for all i ∈ N .By Proposition 2.2, we get for the sequence { u i } i ∈ N strong convergence in L ( R n ),and by the Sobolev embedding, also boundedness in L n/ ( n − ( R n ). We then notethat for every function v ∈ BV( R n ) and every s ∈ (0 ,
1) the interpolation[ v ] W s, ( R n ) ≤ − s nω n s (1 − s ) k v k − sL ( R n ) (cid:0) | Dv | ( R n ) (cid:1) s , is available, see [4, Proposition 4.2]. Therefore, we get strong convergence of { u i } i ∈ N in W s, ( R n ), for every fixed s ∈ (0 , p ′ chosen suchthat 1 < p ′ < nn − n + s − n + s < nn − s ′ defined such that s ′ p ′ = s − ( n − n + s )( p ′ − ⇔ p ′ = nn − s − s ′ p ′ n + s + 1 n + s ′ p ′ n + s , we have 0 < s ′ p ′ < s . We then note that the application of H¨older’s inequality shows[ v ] p ′ W s ′ ,p ′ ( B (0 ,R )) = Z B (0 ,R ) Z B (0 ,R ) | v ( x ) − v ( z ) | p ′ | x − z | n + s ′ p ′ d L n ( x ) d L n ( z ) ≤ (cid:18) Z B (0 ,R ) Z B (0 ,R ) | v ( x ) − v ( z ) | nn − d L n ( x ) d L n ( z ) (cid:19) s − s ′ p ′ n + s × (cid:18) Z B (0 ,R ) Z B (0 ,R ) | v ( x ) − v ( z ) || x − z | n + s d L n ( x ) d L n ( z ) (cid:19) n + s ′ p ′ n + s ≤ (cid:2) L n ( B (0 , R )) (cid:3) s − s ′ p ′ n + s k v k nn − s − s ′ p ′ n + s L n/ ( n − ( B (0 ,R )) [ v ] n + s ′ p ′ n + s W s, ( B (0 ,R )) = 2 (cid:2) L n ( B (0 , R )) (cid:3) ( n − p ′ − k v k n ( p ′ − L n/ ( n − ( B (0 ,R )) [ v ] n − ( n − p ′ W s, ( B (0 ,R )) for every function v ∈ W s, ( B (0 , R )) ∩ L n/ ( n − ( B (0 , R )). Therefore, we also havestrong convergence of the sequence { u i } i ∈ N in W s ′ ,p ′ ( R n ), with s ′ , p ′ chosen as aboveand fixed s ∈ (0 , s ∈ (0 ,
1) and with s ′ p ′ ր s for p ′ ց { s ℓ } ℓ ∈ N , { s ′ ℓ } ℓ ∈ N and { p ′ ℓ } ℓ ∈ N satisfying s ℓ = 1 − /ℓ and s ′ ℓ p ′ ℓ = 1 − /ℓ for each ℓ ∈ N . With Corollary 3.3,we can therefore select with a diagonal argument a subsequence (not relabeled) NOTE ON THE WEAK* AND POINTWISE CONVERGENCE OF BV FUNCTIONS 11 such that pointwise convergence e u i ( x ) → e u ( x ) holds outside of a set E ℓ ⊂ R n withcap s ′ ℓ ,p ′ ℓ ( E ℓ ) = 0, for each ℓ ∈ N . Setting E := T ℓ ∈ N E ℓ , we then have pointwiseconvergence e u i ( x ) → e u ( x ) for every x ∈ R n \ E , while Proposition 2.1 implies that H d ( E ) ≤ H d ( E ℓ ) = 0, for every d > n − ℓ > / ( d − n + 1), which shows thatthe Hausdorff dimension of E is at most n − (cid:3) Pointwise convergence w.r.t. diffuse measures
In this section we first prove, via the slicing technique, that each bounded sequencein BV( R n ) admits a subsequence converging outside of a ν -vanishing set, where ν is a generalized product measure of diffuse type. We then obtain Theorem 1.2 as adirect consequence. For convenience of notation and reference to the literature wehere prefer to always work with the precise representatives, even though the mainarguments exclude jump points of the sequence and the limit function so that theLebesgue representatives would be suitable as well. Theorem 4.1.
Let u ∈ BV( R n ) . Let { u i } i ∈ N be a sequence in BV( R n ) for which {| Du i | ( R n ) } i ∈ N is bounded, and suppose that u ∗ i ( x ) → u ∗ ( x ) for L n -almost every x ∈ R n . Let ν be a positive measure of finite mass admitting a representation ν = L n − ⊗ ν y with ν y ( { t } ) = 0 for all t ∈ R , for L n − -almost every y ∈ R n − . Then for asubsequence ( not relabeled ) we have u ∗ i ( x ) → u ∗ ( x ) for ν -almost every x ∈ R n .Proof. We divide the proof into two steps.
Step 1.
First we consider the one-dimensional case. Let ν be a positive measureof finite mass on R such that ν ( { x } ) = 0 for every x ∈ R . Take u i , u ∈ BV( R ), i ∈ N , with u ∗ i ( x ) → u ∗ ( x ) for L -almost every x ∈ R . (Here we do not assume that {| Du i | ( R ) } i ∈ N is bounded). We consider the set M := n x ∈ R : x / ∈ (cid:16) J u ∪ [ i ∈ N J u i (cid:17) with u ∗ i ( x ) → u ∗ ( x ) as i → ∞ ,ν (( x, ∞ )) , ν (( −∞ , x )) ≤ (2 / ν ( R ) o and observe that M is non-empty, as the convergence u ∗ i ( x ) → u ∗ ( x ) takes place L -almost everywhere, the approximate jump sets J u , J u i are at most countable, andthe conditions ν (( x, ∞ )) , ν (( −∞ , x )) ≤ (2 / ν ( R ) are satisfied on a non-emptyinterval in R (as ν does not charge singletons). Note that we could also work with aconvenient countable and dense subset G ⊂ R as admissible points for this splittingprocedure, in the sense that we could choose it such that for all x ∈ G there hold x / ∈ ( J u ∪ S i ∈ N J u i ) and u ∗ i ( x ) → u ∗ ( x ) as i → ∞ , and then consider instead of M the non-empty set of points x ∈ G such that ν (( x, ∞ )) , ν (( −∞ , x )) ≤ (2 / ν ( R )are satisfied (as done later in Step 2a).We now pick an arbitrary point x ∈ M . Next we split each of the intervals( −∞ , x ) and ( x , ∞ ) into two parts as above, by considering accordingly definedsets M ⊂ ( −∞ , x ), M ⊂ ( x , ∞ ), picking arbitrary points x ∈ M , x ∈ M , andkeeping x := x . Continuing like this, we get a monotonously increasing sequence of collections of points { x ℓj } ℓ − j =1 ⊂ R \ (cid:16) J u ∪ [ i ∈ N J u i (cid:17) , ℓ ∈ N , with (denote x ℓ = −∞ and x ℓ ℓ = ∞ )(4.1) max j ∈{ ,..., ℓ − } ν (( x ℓj , x ℓj +1 )) ≤ (2 / ℓ ν ( R )and u ∗ i ( x ℓj ) → u ∗ ( x ℓj ) as i → ∞ , for each ℓ ∈ N and j = 1 , . . . , ℓ − α ℓi := max j ∈{ ,..., ℓ − } | u ∗ i ( x ℓj ) − u ∗ ( x ℓj ) | so that for any fixed ℓ ∈ N , we have α ℓi → i → ∞ . Note that since necessarily u ∗ ( x ) → u ∗ i ( x ) → x → −∞ , we interpret | u ∗ i ( x ℓ ) − u ∗ ( x ℓ ) | = 0, where x ℓ = −∞ . We have for every ℓ ∈ N Z R | u ∗ i − u ∗ | dν = ℓ − X j =0 Z x ℓj +1 x ℓj | u ∗ i − u ∗ | dν ≤ ℓ − X j =0 ν (( x ℓj , x ℓj +1 )) (cid:16) | u ∗ i ( x ℓj ) − u ∗ ( x ℓj ) | + | D ( u i − u ) | (( x ℓj , x ℓj +1 )) (cid:17) by (2.2) ≤ ℓ − X j =0 ν (( x ℓj , x ℓj +1 )) (cid:16) α ℓi + | D ( u i − u ) | (( x ℓj , x ℓj +1 )) (cid:17) ≤ ℓ − X j =0 ν (( x ℓj , x ℓj +1 )) α ℓi + (2 / ℓ ν ( R ) ℓ − X j =0 | D ( u i − u ) | (( x ℓj , x ℓj +1 )) by (4.1) ≤ ν ( R ) α ℓi + (2 / ℓ ν ( R ) ( | Du i | ( R ) + | Du | ( R )) . (4.2)We will use this estimate to prove the general case. Step 2.
Now we consider the general case. Let { u i } i ∈ N , u be as given in the statementof the theorem.For L n − -almost every y ∈ R n − we have that u ny , ( u i ) ny ∈ BV( R ) for i ∈ N —recall the notation from Section 2.5. For simplicity, we will discard the superscript n and write simply u y , ( u i ) y .We have u ∗ i ( x ) → u ∗ ( x ) for L n -almost every x ∈ R n , implying that for L n − -almost every y ∈ R n − , u ∗ i ( y, t ) → u ∗ ( y, t ) for almost every t ∈ R . In view of (2.5)this implies that for L n − -almost every y ∈ R n − , (( u i ) y ) ∗ ( t ) → ( u y ) ∗ ( t ) for almostevery t ∈ R . Thus for L n − -almost every y ∈ R n − , the functions ( u i ) y , u y satisfythe assumptions for the application of Step 1. Step 2a.
We next reason that the collection of points { t ℓj ( y ) } selected in Step 1 canbe chosen to be L n − -measurable with respect to y ∈ R n − . For this purpose we NOTE ON THE WEAK* AND POINTWISE CONVERGENCE OF BV FUNCTIONS 13 observe that the set n x ∈ R n : x / ∈ (cid:16) J u ∪ [ i ∈ N J u i (cid:17) with u ∗ i ( x ) → u ∗ ( x ) as i → ∞ o is Borel and its complement is of vanishing L n -measure. Hence, by Fubini, we canselect a countable and dense subset G = { g , g , g , . . . } ⊂ R of points and an L n − -negligible set B ⊂ R n − such that ( u i ) y , u y ∈ BV( R ) , for all i ∈ N , (( u i ) ∗ ) y ( t ) = (( u i ) y ) ∗ ( t ) , ( u ∗ ) y ( t ) = ( u y ) ∗ ( t ) , for all i ∈ N , (( u i ) y ) ∗ ( t ) → ( u y ) ∗ ( t ) as i → ∞ , ( y, t ) / ∈ J u ∪ S i ∈ N J u i and so t / ∈ J u y ∪ S i ∈ N J ( u i ) y for all y ∈ R n − \ B and t ∈ G . We next consider the sets B j := n y ∈ R n − : y / ∈ B i for all i ∈ { , , . . . , j − } ,ν y (( g j , ∞ )) , ν y (( −∞ , g j )) ≤ (2 / ν y ( R ) o , which are L n − -measurable in R n − by weak* L n − -measurability of the mapping y ν y and disjoint by construction. As G was chosen dense in R , we hence havethe decomposition R n − = [ j ∈ N B j (as already commented on in Step 1). If we now define a function t := P j ∈ N g j B j ,then we immediately observe that, as a step function, it is L n − -measurable, andfor each y ∈ R n − \ B the point t ( y ) belongs to the set M y (defined just as the set M in Step 1). Iterating this splitting procedure, we then arrive at a monotonouslyincreasing sequence of collections of points { t ℓj ( y ) } ℓ − j =1 ⊂ R \ (cid:16) J u y ∪ [ i ∈ N J ( u i ) y (cid:17) , ℓ ∈ N , which are L n − -measurable with respect to the variable y . As a consequence, dueto the Borel measurability of the precise representatives, also the function α ℓi ( y ) := max j ∈{ ,..., ℓ − } | (( u i ) y ) ∗ ( t ℓj ( y )) − ( u y ) ∗ ( t ℓj ( y )) | is L n − -measurable with respect to y . Let us also note that because of the conver-gence (( u i ) y ) ∗ ( t ) → ( u y ) ∗ ( t ) as i → ∞ , for all y ∈ R n − \ B and t ∈ G , we againhave α ℓi ( y ) → i → ∞ , for fixed ℓ ∈ N and all y ∈ R n − \ B . Step 2b.
We next apply the estimate (4.2) from Step 1. This shows that for L n − -almost every y ∈ R n − (those in R n − \ B ) we have Z R | (( u i ) y ) ∗ ( t ) − ( u y ) ∗ ( t ) | dν y ( t ) ≤ ν y ( R ) α ℓi ( y ) + (2 / ℓ ν y ( R ) ( | D ( u i ) y | ( R ) + | Du y | ( R )) . (4.3) Fix ε >
0. We initially note that by definition of ν , ν ( R n ) = Z R n − ν y ( R ) d L n − ( y ) . Therefore, on the one hand, by choosing a constant M ε > Z A ε ν y ( R ) d L n − ( y ) < ε for A ε := { y ∈ R n − : ν y ( R ) > M ε } , and on the other hand the weighted measure ν y ( R ) d L n − is a finite measure on R n − .By Egorov’s theorem, applied for each fixed ℓ ∈ N , the measure ν y ( R ) d L n − and thesequence of measurable functions { y α ℓi ( y ) } i ∈ N converging pointwisely to zero, wecan find a measurable set A εℓ ⊂ R n − with Z A εℓ ν y ( R ) d L n − ( y ) < − ℓ ε and a sequence of positive numbers { ¯ α ℓi } i ∈ N with ¯ α ℓi → i → ∞ and such that α ℓi ( y ) ≤ ¯ α ℓi for all i ∈ N , for all y ∈ R n − \ A εℓ . Let A ε := S ℓ ∈ N A εℓ , with Z A ε ν y ( R ) d L n − ( y ) < ε. Employing (2.5), (4.3) and finally (2.4) for k = n , we then find Z ( R n − \ A ε ) × R | u ∗ i − u ∗ | d ( L n − ⊗ ν y )= Z R n − \ A ε Z R | u ∗ i ( y, t ) − u ∗ ( y, t ) | dν y ( t ) d L n − ( y )= Z R n − \ A ε Z R | (( u i ) y ) ∗ ( t ) − ( u y ) ∗ ( t ) | dν y ( t ) d L n − ( y ) ≤ Z R n − \ A ε h ν y ( R ) α ℓi ( y ) + (2 / ℓ ν y ( R ) (cid:0) | D ( u i ) y | ( R ) + | Du y | ( R ) (cid:1)i d L n − ( y ) ≤ ¯ α ℓi Z R n − ν y ( R ) d L n − ( y )+ (2 / ℓ sup y ∈ R n − \ A ε ν y ( R ) Z R n − [ | D ( u i ) y | ( R ) + | Du y | ( R )] d L n − ( y ) ≤ ¯ α ℓi ν ( R n ) + (2 / ℓ M ε ( | D n u i | ( R n ) + | D n u | ( R n )) . (4.4)Thus, we obtainlim sup i →∞ Z ( R n − \ A ε ) × R | u ∗ i − u ∗ | d ( L n − ⊗ ν y ) ≤ (2 / ℓ M ε lim sup i →∞ ( | D n u i | ( R n ) + | D n u | ( R n )) ≤ (2 / ℓ M ε lim sup i →∞ ( | Du i | ( R n ) + | Du | ( R n )) → ℓ → ∞ , NOTE ON THE WEAK* AND POINTWISE CONVERGENCE OF BV FUNCTIONS 15 since {| Du i | ( R n ) } i ∈ N is a bounded sequence. By passing to a subsequence (notrelabeled), we have u ∗ i ( x ) → u ∗ ( x ) for L n − ⊗ ν y -almost every x ∈ ( R n − \ A ε ) × R .We can do this for sets A ε = A /j with Z A /j ν y ( R ) d L n − ( y ) < j , for j ∈ N , and by a diagonal argument we obtain that for any j ∈ N , u ∗ i ( x ) → u ∗ ( x )for L n − ⊗ ν y -almost every x ∈ ( R n − \ A /j ) × R , that is, u ∗ i ( x ) → u ∗ ( x ) for L n − ⊗ ν y -almost every x ∈ R n , i.e. ν -almost every x ∈ R n . (cid:3) Proof of Theorem 1.2.
By (2.4), we have | D dn w | = L n − ⊗ | D d w ny | , where | D d w ny | ( { t } ) = 0 for every t ∈ R , for L n − -almost every y ∈ R n − . Thus, byTheorem 4.1 we find a subsequence of { u i } i ∈ N (not relabeled) such that u ∗ i ( x ) → u ∗ ( x ) for | D dn w | -almost every x ∈ R n . Passing to further subsequences (not rela-beled), we then we obtain (after a change of coordinates) for every k = 1 , . . . , n that u ∗ i ( x ) → u ∗ ( x ) for | D dk w | -almost every x ∈ R n . Noting that | D d w | ≤ n X k =1 | D dk w | , we hence have shown the pointwise convergence u ∗ i ( x ) → u ∗ ( x ) for | D d w | -almostevery x ∈ R n . (cid:3) Remarks and examples
In this section we give some remarks concerning Theorems 1.1 and 1.2 and exam-ine their sharpness.
Remark 5.1. If { u i } i ∈ N is a sequence in BV( R n ) with u i → u weakly* in BV( R n ),then it is also norm-bounded in BV( R n ) (see e.g. [2, Proposition 3.13]), and ofcourse for a subsequence we have u ∗ i ( x ) → u ∗ ( x ) for L n -almost every x ∈ R n . Thus,the assumptions of Theorems 1.1, 1.2, and 4.1 are satisfied.In Theorems 1.1 and 1.2, pointwise convergence u ∗ i → u ∗ is stated outside of anexceptional set E ⊂ R n that satisfies dim H ( E ) ≤ n − | D d w | ( E ) = 0, respec-tively. Apart from some specific situations, this cannot be improved to H n − ( E ) = 0or | Dw | ( E ) = 0, meaning that we in particular need to exclude the jump part D j w . Example 5.2.
Let w = u := [0 , ∈ BV( R ) and define u i ( x ) := max { , min { , / ix }} ( −∞ , ( x ) , i ∈ N . Then it is easy to see that { u i } i ∈ N is a norm-bounded sequence in BV( R ) with u i → u weakly* and even strictly in BV( R ). However, we have u ∗ i (0) ≡ / / u ∗ (0).Moreover, u + (0) = 1 and u − (0) = 0, so u ∗ i (0) does not converge to these either.Here | D j u | ( { } ) = H ( { } ) = 1.On the other hand, for any dimension n ∈ N and in the special case that thefunctions u i are defined as convolutions of a function u ∈ BV( R n ) (with standardmollifiers), we have u i → u strictly in BV( R n ) and u ∗ i ( x ) → u ∗ ( x ) for H n − -almostevery x ∈ R n and thus | Du | -almost every x ∈ R n , see [2, Theorem 3.9 & Corollary3.80]. In Theorems 1.1 and 1.2 the pointwise convergence occurs outside of an ( n − | D d w | -negligible set, respectively, but it is not clear how largeexactly such exceptional sets can be. In this regard, consider the following Example 5.3.
Let { q j } j ∈ N be an enumeration of the rational points on the realline, and define E ij := ( q j − /i, q j + 1 /i ) and u i ( x ) := X j ∈ N − j E ij ( x ) , x ∈ R , i ∈ N . Then clearly u i ց u : ≡ L -almost everywhere, and | Du i | ( R ) ≤ X j ∈ N − j | D E ij | ( R ) = X j ∈ N − j +1 = 2 . Thus, { u i } i ∈ N is a norm-bounded sequence in BV( R ) and the assumptions of The-orems 1.1 and 1.2 are satisfied. However, u ∗ i ( q j ) ≥ − j u ∗ ( q j ) as i → ∞ , forevery j ∈ N . Thus, the pointwise convergence u i → u (for any subsequence) failsin a fairly large set, though this set is still σ -finite with respect to the Hausdorffmeasure H n − = H . It is not clear whether the exceptional can in some cases belarger than this. In Section 6 we will show that it is always at most σ -finite withrespect to H n − in two special cases: when n = 1 and when { u i } i ∈ N is a decreasingsequence.One can also ask whether it is necessary to pass to a subsequence in Theorems 1.1and 1.2; the answer is in general yes. Example 5.4.
Let C = T k ∈ N C k ⊂ [0 ,
1] be the standard 1 / C = [0 ,
1] and C k is obtained iteratively from C k − by removing the open middlethird of each interval, meaning that in the end C k consists of 2 k compact intervals C k , . . . , C k k , each of length 3 − k . Note that dim H ( C ) = log (2). Let { E i } i ∈ N be thesequence of sets C , C , C , C , C , C , C , . . . . Let v be the Cantor-Vitali function, let w = u := v [0 , ∈ BV( R ), and define u i := u + E i , i ∈ N . Then u i → u in L ( R ) and L -almost everywhere, and | Du i | ( R ) = 4 for all i ∈ N ,hence, { u i } i ∈ N is a norm-bounded sequence in BV( R ). However, for all x ∈ C , andthus for all x in the support of | D c u | , we have that x ∈ E i for infinitely many i ∈ N .For such i ∈ N we have u ∗ i ( x ) ≥ u ∗ ( x ) + 1 / u ∗ i ( x ) fails to converge to u ∗ ( x ). Hence, it is necessary to pass to a subsequence in order to obtain pointwiseconvergence outside of ( n − | D d u | -negligible sets.Theorems 1.1 and 1.2 both involve an exceptional set E where the precise rep-resentatives (for a subsequence) do not converge, but they are different in nature.Theorem 1.1 gives the upper bound n − E and henceneglects ( n − E is in particular | D c u | -vanishing, and here D c u can be supported on an ( n − Example 5.5.
Let C = T k ∈ N C k ⊂ [0 ,
1] be a generalized Cantor set, where C =[0 ,
1] and C k is obtained from C k − by removing an open set in the middle of eachinterval of fraction 1 − · − k − , meaning that in the end C k consists of 2 k compact NOTE ON THE WEAK* AND POINTWISE CONVERGENCE OF BV FUNCTIONS 17 intervals of length 3 − k . This generalized Cantor set is uncountable by construction,exactly as the standard 1 / d >
0, the Hausdorff pre-measure of fineness δ > H dδ ( C ) = inf n X j ∈ N (diam U j ) d : C ⊂ [ j ∈ N U j , diam U j < δ for all j ∈ N o ≤ k [3 − k ] d k →∞ −→ , by taking the intervals from C k as an admissible covering of C , for k > − log δ . Thisshows, for each fixed d >
0, that H d ( C ) = 0, implying dim H ( C ) = 0. Introducingfor each k ∈ N the piecewise affine and monotone functions u k ( x ) = 2 − k k Z x C k ( t ) dt for x ∈ [0 , , we easily verify thatmax x ∈ [0 , | u k +1 ( x ) − u k ( x ) | = 2 − k − (1 − · − k − ) < − k − . Thus, { u i } i ∈ N is a Cauchy sequence in C ([0 , u ∈ C ([0 , u ∈ BV(0 , u k is constant on each connectedcomponent of [0 , \ C (for k sufficiently large), we finally conclude that Du is con-centrated on the 0-dimensional set C and that it is purely Cantor, i.e. Du = D c u .Note that we have the following consequence of the proof of Theorem 4.1 in theone-dimensional setting. Proposition 5.6.
Let u ∈ BV( R ) . Let { u i } i ∈ N be a sequence in BV( R ) for which {| Du i | ( R ) } i ∈ N is bounded, and suppose that u ∗ i ( x ) → u ∗ ( x ) for L -almost every x ∈ R n . Let w ∈ BV( R ) . Then we have Z R | u ∗ i − u ∗ | d | D d w | → as i → ∞ . Proof.
Equation (4.2) with ν = | D d w | giveslim sup i →∞ Z R | u ∗ i − u ∗ | d | D d w |≤ lim sup i →∞ (cid:0) | D d w | ( R ) α ℓi + (2 / ℓ | D d w | ( R )( | Du i | ( R ) + | Du | ( R )) (cid:1) = (2 / ℓ | D d w | ( R ) lim sup i →∞ (cid:0) | Du i | ( R ) + | Du | ( R ) (cid:1) , which becomes arbitrarily small as ℓ → ∞ . (cid:3) Example 5.7.
In general dimensions n ≥ Z R n | u ∗ i − u ∗ | d | D d u | → i → ∞ , even if u i → u strongly in BV( R n ), since u ∗ need not be integrable with respectto | D d u | . This can be seen by considering u ( x ) := η ( x ) | x | − / in R , where η is a smooth function with B (0 , ≤ η ≤ B (0 , . Then for u i := min { u, i } , i ∈ N , wehave u i → u in BV( R ) but Z R | u ∗ i − u ∗ | d | D d u | = π Z i − ( r − / − i ) r − / dr = ∞ for all i ∈ N . Example 5.8.
The motivation for this paper arose from the theory of liftings ,see [14]. Let Ω ⊂ R n be an open, bounded set. A lifting is a measure γ ∈ M (Ω × R N ; R N × n ) for which there exists a function u ∈ [BV(Ω)] N with integral average 0and such that the chain rule formula Z Ω ∇ x ϕ ( x, u ( x )) d L n ( x )+ Z Ω × R N ∇ z ϕ ( x, z ) dγ ( x, z ) = 0 for all ϕ ∈ C (Ω × R N )holds. Here u =: [ γ ] can be shown to be unique. Given a function u ∈ [BV(Ω)] N with integral average 0, an elementary lifting γ [ u ] is defined by h ϕ, γ [ u ] i := Z Ω Z ϕ ( x, u θ ( x )) dθ dDu ( x ) for all ϕ ∈ C (Ω × R N ) , where u θ ( x ) := e u ( x ) for x ∈ R n \ S u and u θ ( x ) := θu − ( x ) + (1 − θ ) u + ( x ) for x ∈ J u . The family of approximable liftings is then defined as the weak* limits in M (Ω × R N ; R N × n ) of sequences of elementary liftings. Thus, for an approximable lifting γ we have a sequence { u i } i ∈ N in [BV(Ω)] N with integral averages 0 and γ [ u i ] ∗ ⇀ γ in M (Ω × R N ; R N × n ), and then we also have u i ∗ ⇀ u := [ γ ] in [BV(Ω)] N . Therefore,it is of interest to better understand weak* convergence in the BV space, and weexpect that the results of the current paper may be of use in further research on(approximable) liftings. In particular, Theorem 1.2 may be of help in investigatingdecomposition of approximable liftings into mutually singular measures, which arerelated to the measures D a u , D c u , D s u ; see [14, Theorem 3.11] for an existingstructure theorem. 6. Two special cases
In this section we show that we can obtain pointwise convergence outside of anexceptional set with σ -finite H n − -measure in two special cases: when n = 1, andwhen { u i } i ∈ N is a decreasing sequence. We start with the first case. Proposition 6.1.
Let u ∈ BV( R ) . Let { u i } i ∈ N be a sequence in BV( R ) for which {| Du i | ( R ) } i ∈ N is bounded, and suppose that u ∗ i ( x ) → u ∗ ( x ) for L -almost every x ∈ R . Then there exists an at most countable set E ⊂ R such that for a subsequence ( not relabeled ) we have u ∗ i ( x ) → u ∗ ( x ) for every x ∈ R \ E . Remark 6.2.
Note that this proposition may appear to be an improvement overStep 1 of the proof of Theorem 4.1. However, unlike in Step 1, in this propositionwe already need to pass to a subsequence. Therefore, it is not clear how one would
NOTE ON THE WEAK* AND POINTWISE CONVERGENCE OF BV FUNCTIONS 19 apply this proposition in Step 2, since the subsequence could be different for eachone-dimensional section ( u i ) y . Proof.
Since the sequence of measures { Du i } i ∈ N is mass-bounded, passing to a subse-quence (not relabeled) we find a positive finite measure ν on R such that | Du i | ∗ ⇀ ν .Note that ν ≥ | Du | (see [2, Proposition 1.62]). Let E be the set of singletons chargedby ν ; then E is at most countable, so that we can write E = { a , a , . . . } . Let M ∈ N and define α M := P k>M ν ( a k ). Fix an arbitrary compact set K ⊂ R \ { a , . . . , a M } and ε >
0. Denote µ := ν | R \ E , so that µ does not charge singletons.By a similar splitting procedure as in Step 1 of the proof of Theorem 4.1, we findclosed intervals I j ⊂ R \ { a , . . . , a M } , j = 1 , . . . , L ∈ N , such that K ⊂ L [ j =1 I j and µ ( I j ) < ε for each j = 1 , . . . , L, and, denoting by x j the left end point of the interval I j , such that x j / ∈ J u ∪ S i ∈ N J u i and | u ∗ i ( x j ) − u ∗ ( x j ) | → i → ∞ , for each j = 1 , . . . , L . Then ν ( I j ) ≤ α M + µ ( I j ) ≤ α M + ε for each j = 1 , . . . , L . By the weak* convergence | Du i | ∗ ⇀ ν , we havelim sup i →∞ | Du i | ( I j ) ≤ ν ( I j )for each j = 1 , . . . , L . Thus, for every x ∈ I j , recalling (2.3) and | Du | ≤ ν , we getlim sup i →∞ | u ∗ i ( x ) − u ∗ ( x ) | ≤ lim sup i →∞ (cid:0) | u ∗ i ( x j ) − u ∗ ( x j ) | + | D ( u i − u ) | ( I j ) (cid:1) ≤ lim sup i →∞ | Du i | ( I j ) + | Du | ( I j ) ≤ ν ( I j ) + ν ( I j ) ≤ α M + 2 ε. Letting ε → R \ { a , . . . , a M } with compact subsets K , weget lim sup i →∞ | u ∗ i ( x ) − u ∗ ( x ) | ≤ α M for all x ∈ R \ { a , . . . , a M } , Thus, letting M → ∞ , we finally end up withlim sup i →∞ | u ∗ i ( x ) − u ∗ ( x ) | = 0for all x ∈ R \ E , which completes the proof of the proposition. (cid:3) We next examine the second case, where we deal with a decreasing sequence.
Proposition 6.3.
Let u ∈ BV( R n ) . Let { u i } i ∈ N be a decreasing sequence in BV( R n ) for which {| Du i | ( R n ) } i ∈ N is bounded, and suppose that u i ( x ) → u ( x ) for L n -almostevery x ∈ R n . Then there exists a set E ⊂ R n such that E is σ -finite with respectto H n − and u ∗ i ( x ) → u ∗ ( x ) for every x ∈ R n \ E . Proof.
By Proposition 2.2, we have u i → u in L ( R n ), and we can in fact assume u i → u in L ( R n ) (otherwise, we multiply the sequence by cut-off functions η R with B (0 ,R/ ≤ η R ≤ B (0 ,R ) for R ∈ N ).First we assume that u ≡
0. Let P := [ i ∈ N (cid:0) S u i \ J u i (cid:1) , so that H n − ( P ) = 0, and let E ⊂ R n be the set where the convergence u ∗ i → u ∗ fails. Then E = S ∞ k =1 E k with E k := { x ∈ R n : lim i →∞ u ∗ i ( x ) ≥ /k } . Fix k ∈ N . By the coarea formula (2.1), we have for every i ∈ N (6.1) Z /k /k | D { u i >t } | ( R n ) dt ≤ Z ∞−∞ | D { u i >t } | ( R n ) dt = | Du i | ( R n ) . Thus, for every i ∈ N we can choose t i ∈ (1 /k, /k ) such that | D { u ∗ i >t i } | ( R n ) ≤ k | Du i | ( R n ) . Now, setting S i := { x ∈ R n : u ∗ i ( x ) > t i } , we have S i → L ( R n ) and(6.2) lim sup i →∞ | D S i | ( R n ) ≤ k lim sup i →∞ | Du i | ( R n ) < ∞ . Fix i ∈ N and let x ∈ E k \ P . Then, by the fact that { u i } i ∈ N is a decreasing sequence,we have x ∈ S i . Using the fact that x is either a Lebesgue or a jump point for u i by definition of the set P , we can further verifylim r → L n ( B ( x, r ) ∩ S i ) L n ( B ( x, r )) ≥ . Setting R i := ω − /nn (3 L n ( S i )) /n , for all r ≥ R i we have L n ( B ( x, r ) ∩ S i ) L n ( B ( x, r )) ≤ . Thus, by continuity we find 0 < r x ≤ R i such that L n ( B ( x, r x ) ∩ S i ) L n ( B ( x, r x )) = 13 . By the relative isoperimetric inequality (with constant C P depending only on n ),cp. [2, Remark 3.45], we have(6.3) L n ( B ( x, r x )) r x ≤ C P | D S i | ( B ( x, r x )) . The collection { B ( x, r x ) } x ∈ E k \ P is a covering of E k \ P . By the Vitali 5-coveringtheorem, we then find a countable collection of disjoint balls { B ( x j , r j ) } j ∈ N suchthat the balls { B ( x j , r j ) } j ∈ N cover E k \ P . Thus, using (6.3), we find for the( n − R i of the set E k \ P theestimate H n − R i ( E k \ P ) ≤ ω n − X j ∈ N (5 r j ) n − ≤ C P n − ω n − ω n X j ∈ N | D S i | ( B ( x j , r j )) ≤ C ( n ) | D S i | ( R n ) . NOTE ON THE WEAK* AND POINTWISE CONVERGENCE OF BV FUNCTIONS 21
Letting i → ∞ , so that also R i →
0, by (6.2) we get H n − ( E k \ P ) ≤ C ( n ) lim sup i →∞ | D S i | ( R n ) < ∞ . Since this holds for each k ∈ N , we obtain that the set E \ P = S k ∈ N E k \ P is σ -finitewith respect to H n − , and then so is E . Note that for every x ∈ R n \ E k , we havelim i →∞ u ∗ i ( x ) < /k . Thus, for every x ∈ R n \ E , we have lim i →∞ u ∗ i ( x ) = 0 = u ∗ ( x ),completing the proof in the case u ≡ { u i − u } i ∈ N is a decreasing sequence in BV( R n ) for which {| D ( u i − u ) | ( R n ) } i ∈ N is bounded, and u i − u → L n -almost everywhere. Bythe first part, we have ( u i − u ) ∗ → σ -finite with respectto H n − . Note that outside of the H n − -negligible set( S u \ J u ) ∪ [ i ∈ N (cid:0) S u i \ J u i (cid:1) we have ( u i − u ) ∗ = u ∗ i − u ∗ . Therefore, the assertion of the proposition follows. (cid:3) Acknowledgments.
The authors wish to thank Giles Shaw for posing the questionthat led to this research and for discussions on the topic, and also Jan Kristensenand Bernd Schmidt for discussions.
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L. B.: Institut f¨ur Mathematik, Universit¨at Augsburg, Universit¨atsstr. 14, 86159 Augs-burg, Germany
E-mail address : [email protected] P. L.: Institut f¨ur Mathematik, Universit¨at Augsburg, Universit¨atsstr. 14, 86159 Augs-burg, GermanyAcademy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing100190, PR China
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