Some Fine Properties of BV Functions on Wiener Spaces

Abstract In this paper we define jump set and approximate limits for BV functions on Wiener spaces and show that the weak gradient admits a decomposition similar to the finite dimensional case. We also define the SBV class of functions of special bounded variation and give a characterisation of SBV via a chain rule and a closure theorem. We also provide a characterisation of BV functions in terms of the short-time behaviour of the Ornstein-Uhlenbeck semigroup following an approach due to Ledoux.


Introduction
In this paper we continue the investigation of the properties of functions with bounded variation in in nite dimensional spaces, that is the setting of abstract Wiener spaces. The theory started with the papers [14], [15] where essentially a probabilistic approach was given and has been subsequently developed in [17], [5] with a more analytic approach.
Motivations for considering functions with bounded variation come from stochastic analysis, see e.g. [18], [28], [24], [25]; recently, in [23], properties of sets with nite Gaussian perimeter have been linked to some application in information technology. We point out also [26], for an application of BV functions to Lagrangian ows in Wiener spaces.
The aim of this paper is twofold; in Section 3 we give an equivalent characterisation of functions with bounded variation following an approach suggested by Ledoux in [19] and subsequently generalised in Euclidean spaces in [22]. In addition, in Section 4, following [6], [2], [3], we discuss the decomposition of the gradient of a BV function into absolutely continuous, jump and Cantor part.
We close the paper by introducing the notion of special function of bounded variation: the de nition coincides with the original one given by De Giorgi and Ambrosio in [11]. We also give the characterisation based on the chain rule proposed by Alberti and Mantegazza [1]; such characterisation turns out to be very useful when giving closure and compactness results. We point out that for the compactness theorem, the only di erence with respect to the Euclidean case, is that we have to assume a priori some convergence of the sequence, for instance the convergence in measure, from which we deduce the separate weak convergence of the two parts of the total variation measure. Given a set E ⊂ X with nite perimeter, we deduce from this result the compactness w.r.t. the weak topology of L p (E, γ) of bounded and closed subsets of the Sobolev H ,p (E, γ), < p < ∞. This Sobolev space, de ned in (5.4) below, consists of all L p (E, γ) functions whose null extension which is a dense (even w.r.t. to the Hilbertian norm) subspace of H. For h = x * ∈ X * , the correspondingĥ is precisely the map x → x, x * . A relevant role in the sequel is played by the Ornstein-Uhlenbeck semigroup T t , pointwise de ned for u ∈ L (X, γ) by Mehler's formula Given an n-dimensional subspace F ⊂ X * , we frequently consider an orthonormal basis {h , . . . , hn} of F and the factorization X = F ⊕ ker(π F ) , π F is the continuous linear map The decomposition x = π F (x) + (x − π F (x)) is well de ned because π F • π F = π F and so x − π F (x) ∈ ker(π F ); in turn, this follows fromĥ i (h j ) = δ ij . Thanks to the fact that |h i | H = , this induces a factorization γ = γ F ⊗ γ ⊥ F , with γ F the standard Gaussian in F (endowed with the metric inherited from H) and γ ⊥ F Gaussian in ker(π F ) with Cameron-Martin space F ⊥ . Let us de ne the space of functions of bounded variation in X. First, let us recall the de nition of the Orlicz space L log / L(X, γ): L log / L(X, γ) := u : X → R measurable : A / (λ|u|) ∈ L (X, γ) for some λ > , endowed with the Luxemburg norm ||u|| L log / L(X,γ) := inf λ > : Given h ∈ H and f ∈ C b (X), beside the directional derivative of f along h, denoted ∂ h f , we de ne the formal adjoint di erential operator ∂ * h f = ∂ h f −ĥf and, for φ ∈ FC b (X, H), we de ne the divergence as follows: If in particular u = χ E is the characteristic function of a measurable set E and u ∈ BV(X, γ) we say that E has nite perimeter and set P H (E, ·) = |D H u|(·).
The study of BV(X, γ) functions has been mainly developed so far for nite perimeter sets, see [17], [6], [2], [3], [9] and the rst question that has been addressed is the identi cation of the subset of the topological boundary of E where the perimeter measure is concentrated. It is known that Let us explain the meaning of the above symbols. For an n-dimensional subspace F ⊂ X * as before, and for y ∈ Ker(π F ), we denote by the section of B ⊂ X. Moreover, denoting by Gn(z) := ( π) −n/ exp(−|z| / ) the n-dimensional Gaussian kernel, we take advantage of the above described decomposition and de ne the pre-Hausdor measures in X induced by F setting Fixing an increasing family F = {Fn} n≥ of nite-dimensional subspaces of X * , whose union is dense in H, we de ne the (∞ − )-dimensional spherical Hausdor measures S ∞− F , basically introduced in [13], see also [17], [6], by setting: In the same vein, if E is a set with nite perimeter, we de ne the cylindrical essential boundary ∂ * F E in the rst equality in (2.2), by where, with the usual notation, ∂ * F E := y + z : y ∈ Ker(π F ), z ∈ ∂ * Ey and ∂ * Ey is the essential boundary of the section Ey in nite dimensions. The measure S ∞− is de ned by taking the supremum of S ∞− F when F runs along all the nite dimensional subspaces of X * . For a comparison of the two approaches we refer to [2,Remark 2.6]. The set E / of points of density / is de ned in [2] by using the semigroup T t introduced in (2.1).
We denote by E / the set where we apply the semigroup T t to the Borel representative of the set E, still denoted by E. Notice that the representation in the last term in (2.2) has the advantage of being coordinate-free. Thanks to (2.2), in all the statements that hold up to |D H χ E | negligible sets we may use both representations indi erently. Let us recall the main result of [3]. For h ∈ H, we de ne the halfspace having h as its "inner normal" by Let us draw a consequence that is useful later.

Corollary 2.3. Given two nite perimeter sets E, F, the equality ν E
Proof. If E ⊂ F then this is an easy consequence of Theorem 2.2. Indeed, for S ∞− -a.e. x the rescaled sets Since each halfspace is determined by the normal unit vector we get the thesis. For the general case, notice that Conversely, if there is n ∈ N such that x ∈ ∂ * F k E for all k ≥ n, then, by monotonicity, x ∈ ∂ * F E as well. Therefore, if u ∈ BV(X, γ) and x ∈ X, the set Jx = {t ∈ R : and k ≥ n and x ∈ ∂ * F {u > τ}, which proves that Jx is an interval. We denote by u ∧ (x) ≤ u ∨ (x) the endpoints of Jx.

Functions of bounded variation and short time behaviour of the semigroup
The following characterisation of BV(X, γ) functions in terms of the short-time behaviour of T t is by now well-known: In this Section we present a second way to characterise sets and functions of bounded variation in terms of the semigroup; this approach was suggested, using the heat semigroup, by Ledoux [19] and subsequently investigated in [22]. Even though the results in this section are not necessary in the sequel of this paper, they seem to be worth presenting here, also in view of di erent applications, see e.g. [23]. If E ⊂ X has nite perimeter, then then E has nite perimeter and the limiting formula (3.1) holds. We notice that we can equivalently write let us also de ne the function the characterisation (3.1) of sets with nite perimeter following the Ledoux approach is a consequence of Theorem 3.2 below. In the proof we need some properties of the conditional expectation and of the semigroup that are likely known. We prove them for the convenience of the reader, as we did not nd a reference. We denote by πn : X → R n and Πn = πn × πn : X × X → R n the nite dimensional projections and by Fn and Fn × Fn the induced σ-algebras. By p : X × X → X and p n : R n × R n → R n we denote the projections on the rst components and by R t : Lemma 3.1. Let πn, Πn, Fn and Fn × Fn be as before; then 2) for any f ∈ L (X, γ), In addition, if T n s and T n s denote the Ornstein-Uhlenbeck semigroups on R n and R n respectively, then 3) for any F ∈ L (R n , γ n ), 4) for any f ∈ L (R n , γn) and any (x, y) ∈ R n there holds where indeed the function T n s (f • p n ) depends only on the rst n variables in R n . Then The general statement follows since the sets of the form A × B form a basis for the σ-algebra Fn × Fn.
then u ∈ BV(X, γ). In addition, the following limit holds Proof. The rst part of the proof is based on [2, Lemma 2.3]. We start by considering a function v ∈ C b (R n ); then denoting by γn and γ n the standard Gaussian measures on R n and R n respectively, and using the rotation invariance of the Gaussian measure, that is the fact that R τ γ n = γ n , where Rτ : R n → R n is the map de ned in (3.4) and R τ is the push-forward operator, we obtain the following estimate: Here we have used the equality R n |∇v(x) · y|dγn(y) = |∇v(x)| /π. Notice that taking a sequence of FC b functions that converges in variation, i.e., v k → v in L (X, γ) such that |D H v k |(X) → |D H v|(X), the above estimate holds for every v ∈ BV(X, γ). We now show that Indeed, the linear functionals on C b (R n ) have, thanks to (3.8), norm uniformly bounded by In addition Then the functionals L t weakly * converge to the functional L and If now u ∈ BV(X, γ), we can consider a cylindrical approximation u j = v j • π j , with v j ∈ C b (R n j ) and π j : X → R n j a projection induced by orthonormal elements h , . . . , hn j ∈ H; the cylindrical approximation can be chosen in such a way that which proves the inequality in (3.5).
Let now x u ∈ L log / L(X, γ) and assume (3.6), that we can rewrite as where p : X × X → X is the projection p (x, y) = x and R t is the rotation de ned in (3.4). Then, if Fn is the σ-algebra induced by the projection πn : X → R n and Fn × Fn the σ-algebra induced by Πn : X × X → R n , Πn(x, y) = (πn(x), πn(y)), we have that where we have used the properties (1), (2) of the conditional expectation stated in Lemma 3.1. So we may assume that v ∈ L log / L(R n , γn) is a function such that and we prove that v ∈ BV(R n , γn). If we denote by (T n s )s and (T n s )s the Ornstein-Uhlenbeck semigroups on R n and R n respectively, then since they are mass preserving, we have that Thanks to the previous arguments, we obtain that The same conclusion holds for u ∈ L log / L(X, γ), by taking conditional expectations. The limiting formula follows from the inequalities which is true for any n ∈ N and s > ; the result then follows by letting n → +∞ and s → .

Moreover, equality holds if and only if E is a halfspace with ∈ ∂E.
Proof. Starting from (3.2) and (3.5) we get whence, taking into account that we deduce (3.10) by taking the limit as t → ∞ in (3.11). Let us prove that the only set attaining equality in (3.10) is a halfspace with α = . Indeed, by Ehrhard symmetrisation, if E is a set with nite perimeter, then where E s is a halfspace with γ(E) = γ(E s ); in addition, equality holds in the last estimate if and only if E is equivalent to a halfspace. Then The rst inequality is an equality if and only E is a halfspace; for the second inequality, if we take E s = E h α with |h| H = , then an explicit computation yields We have then shown that F(α) ≥ √ √ π ; a direct computation shows that the only solution of the equation Remark 3.4. For the halfspace E = E h , |h| H = , the following equality holds: Moreover, it is the only set with nite perimeter with this property. First, let us explicitly compute the quantity Using the fact that the γ-measurable linear functional x →ĥ(x) has Gaussian law, by writing χ E (x) = χ (−∞, ) (ĥ(x)), we may write

Decomposition of the gradient and chain rule
In this section we discuss a few ner properties of a function u with bounded variation. If we x h ∈ H, recall that we are denoting by µ h the measure [D H u, h] H de ned as The measure D H u can be decomposed into an absolutely continuous part D a H u with respect to γ, whose density is denoted by ∇ H u, and a singular part D σ H u, as follows In this way the measure µ h admits the Radon-Nikodym decomposition . We recall also that if we write X = F ⊕ F ⊥ with F = span{h} and F ⊥ = kerπ F , then by de ning uy(t) = u(y + th) we have that for γ ⊥ F -a.e. y ∈ F ⊥ , uy ∈ BV(F, γ F ); we de ne We recall the formula (a simple consequence of Fubini's theorem) and we analyse in the next lemma the e ect of the decomposition in absolutely continuous and singular part. and As a consequence, for γ ⊥ F -a.e. y ∈ F ⊥ there holds u y (z) = (∂ h u)y(z) for γ F -a.e. z ∈ F.
Notice that for γ ⊥ F -a.e. y ∈ F ⊥ the measure D σ F uy is singular with respect to γ F , so that we may de ne the γ F -negligible Borel set Since µ σ h (B) = µ σ h (B ∩ A), we deduce that µ σ h ⊥ γ. By (4.2) and the uniqueness of the Radon-Nikodym decomposition we get that µ a h and µ σ h are given by (4.3), (4.4). The fact that u y = (∂ h u)y γ ⊥ F -a.e. for γ ⊥ F a.e. y ∈ F ⊥ is then an easy consequence.

de ne two H-valued measures denoted by D J H u and D C H u.
Proof. Notice that but by the linearity if (B j ) is a countable partition of X such that λ(B j ) ≠ for all j ∈ N, setting k j = It follows that λ(B) ∈ H for any B ∈ B(X), that λ belongs to M(X, H) and the de nition is independent of the basis. Let us show that the set functions de ned in (4.6), (4.7) verify the hypotheses of linearity and continuity with respect to h and also the boundedness assumption (4.5). The linearity of λ J h , λ C h follows basically arguing as in [5,Proposition 4.8], see also [4,Theorem 3.108]. Indeed, take h ∈ F = span{h , h }. Then, by the nite dimensional result. Moreover, the boundedness follows from According to Lemma 4.2, we de ne the jump and the Cantor parts of D H u as because By is a locally nite set for γ ⊥ -a.e. y ∈ h ⊥ and D C h uy(By) = by the analogous one-dimensional property.
Let us consider the jump part. In the following de nition we think of F as a xed increasing sequence of nite dimensional subspaces of X * , as explained in Section 2.

De nition 4.4.
Let u ∈ BV(X, γ) and let D ⊂ R be a countable dense set such that {u > t} has nite perimeter for all t ∈ D. De ne the discontinuity set of u as By de nition, S(u) is σ-nite with respect to S ∞− . Let us show that D J H u is concentrated on S(u).
for every Borel set B. Therefore, for we get |D H u|(Bu) = and the existence ofũ(x) for |D H u|-a.e. x ∈ X \ S(u). Let us prove the uniqueness: if there were s ≠ t ∈ R such that x ∈ ∂ * F {u > t} ∩ ∂ * F {u > s} then the set Jx of such numbers, according to Remark 2.4, would be an interval containing a pair s ≠ t ∈ D, whence we would get the contradiction x ∈ S(u).
According to Lemma 4.6, for |D H u|-a.e. x ∈ X \ S(u) we may de nẽ u(x) = t, (4.13) where t ∈ R is the unique value such that x ∈ ∂ * F {u > t} and we callũ(x) the approximate limit of u at x.
Proof. Let us rst show that v = ψ • u belongs to BV(X, γ). To this end, notice rst that v has at most a linear growth, hence it belongs to L log / L(X, γ). Moreover, if um are the canonical cylindrical approximations of u we have that ψ • um ∈ BV(X, γ), ψ • um → v in L (X, γ) and by lower semicontinuity. Next, we prove that S(v) ⊂ S(u). Indeed, if x ∈ S(v) then there are s, t ∈ D, s < t, such that x ∈ ∂ * F {v > s} ∩ ∂ * F {v > t}. By de nition of cylindrical essential boundary, there are two nite dimensional subspaces F , F ∈ F such that for any G ∈ F containing both F and F we have For every such G we may write x = y + z, with y ∈ ker π G and z ∈ ∂ * {v > s}y ∩ ∂ * {v > t}y. By the nite dimensional case, see [4,Theorem 3.96], z ∈ S(v)y implies z ∈ S(u)y, and therefore x = y + z ∈ S(u). Let B ⊂ S(u) and assume that ψ is increasing. In this case, for any x ∈ S(u) and by the coarea formula we get but then, if we set t = ψ(s), we get {ψ(u) > t} = {u > s}, and thus  Lemma 4.6 and (4.13), and then arguing as before we nd Finally, for ψ ∈ C (R) ∩ Lip(R), if we x L > ||ψ ||∞, we may apply the previous result to the strictly increasing function ψ(t) + Lt.
follows from Theorem 4.5 and the σ-niteness of S(u) with respect to S ∞− . Therefore, if u ∈ SBV(X, γ) the statement is obvious. The opposite implication follows from Lemma 4.3.
As an application of the previous result, we deduce a compactness theorem for the space H ,p (E, γ), in the spirit of [27, Chapter 5, Section 3].
Proof. Let (u k ) be a bounded sequence in H ,p (E, γ). Since a function u belongs to SBV(X, γ) if and only if all its truncations u K = (u ∧ K) ∨ (−K) are SBV, we may suppose that the u k are equibounded. Eventually, a diagonal argument allows us to remove this hypothesis. By the boundedness in L p (E, γ) we infer that a subsequence (which we don't relabel) is weakly converging to a function u in L p (E, γ). Let us show that u ∈ H ,p (E, γ). To this aim, notice rst that by Mazur's lemma a suitable sequence (v k ) of convex combinations of the (u k ) converges strongly to u in L p (E, γ) and that the null extensions v * k still belong to H ,p (E, γ) because, as for the (u * k ), D σ H v * k (X \ ∂ * F E) = . Therefore, we may apply Theorem 5.4 to the sequence (v * k ) ⊂ SBV(X, γ) with ψ(t) = |t| p and θ(t) = and conclude that u * ∈ SBV(X, γ). Finally, since D σ H u * (X \ ∂ * F E) = by the weak convergence of D J v * k and ∇ H u * ∈ L p (E, γ) by (5.3), the proof is complete.