Integral Representation of Linear Functionals on Vector Lattices and ...

Advanced Studies in Pure Mathematics 3?, 200? Stochastic Analysis and Related Topics in Kyoto pp. 1–20

Integral Representation of Linear Functionals on Vector Lattices and its Application to BV Functions on Wiener Space Masanori Hino Abstract. We consider vector lattices D generalizing quasi-regular Dirichlet spaces and give a characterization for bounded linear functionals on D to have a representation by an integral with respect to smooth measures. Applications to BV functions on Wiener space are also discussed.

§1.

Introduction

Let X be a compact Hausdorff space and C(X) the Banach space of all continuous functions on X with supremum norm. The Riesz representation theorem says that every bounded linear operator on C(X) is realized by an integral with respect to a certain finite signed measure on X. As a variant of this fact, Fukushima [7] proved that, for any quasi-regular Dirichlet form (E, F) and for u ∈ F, E(u, ·) is represented as an integral by a smooth signed measure if and only if |E(u, v)| ≤ Ck kvkL∞

for all v ∈ Fb,Fk , k ∈ N

holds for some nest {Fk }k∈N and some constants Ck , k ∈ N. As its applications, he gave a characterization for additive functionals of function type for (E, F) to be semimartingales, and also proved the smoothness of the measures associated with BV functions ([7, 8, 9]). In this paper, we show a corresponding result in the framework of vector lattices generalizing quasi-regular Dirichlet spaces. The proof is similar to that in [7] but based on a purely analytical argument, unlike [7] where probabilistic methods are used together. Typical examples Partially supported by Ministry of Education, Culture, Sports, Science and Technology, Grant-in-Aid for Encouragement of Young Scientists, 15740089, 2003.

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which lie in this framework are first order Sobolev spaces derived from a gradient operator and fractional order Sobolev spaces by a real interpolation method with differentiability index between 0 and 1. Using such results, we can improve the smoothness of the measure associated with BV functions on Wiener space, discussed in [8, 9]. We can find several studies closely related to this article regarding the Riesz representation theorem, e.g., in [22, 14]. Their frameworks are based on Markovian semigroups and the function spaces are derived from their generators, which seems to be suitable for complex interpolation spaces. Ours is based on the lattice property instead and fits for real interpolation spaces. The organization of this paper is as follows. In Section 2, we give a general framework and preparatory lemmas, which are slight modifications of what have been developed already in the case of Dirichlet spaces or in the framework of the nonlinear potential theory. We also give some examples there. In Section 3, the representation theorems are proved. In Section 4, we discuss some applications to BV functions on Wiener space. §2.

Framework and main results

Let X be a topological space and λ a Borel measure on X. Let L0 (X) be the space of all λ-equivalence classes of real-valued Borel measurable functions on X. We will adopt a standard notation to describe function spaces and their norms, such as Lp (X) (or simply Lp ) and k · kLp . We suppose that a subspace D of L0 (X) equipped with norm k · kD satisfies the following. (A1) (D, k · kD ) is a separable and uniformly convex Banach space. (A2) (Consistency condition) If a sequence in D converges to 0 in D, then its certain subsequence converges to 0 λ-a.e. Since D is assumed to be uniformly convex, it is reflexive and the BanachSaks property holds: every bounded sequence in D has a subsequence whose arithmetic means converge strongly in D (see [16, 19, 13] for the proof). The following lemma is proved by a standard argument. Lemma 2.1. Let {fn }n∈N be a sequence bounded in D. Then, there exists a subsequence {fnk }k∈N such that fnk converges to some f weakly Pk in D and the arithmetic means (1/k) j=1 fnj converge to f strongly in D. Moreover, kf kD ≤ lim inf k→∞ kfnk kD . If furthermore fn converges to some g λ-a.e., then it holds that g ∈ D and fn weakly converges to g in D.

Integral representation

3

Proof. By virtue of the Banach-Alaoglu theorem and the reflexivity of D, {fn }n∈N is weakly relatively compact in D. Using the Banach-Saks property together, we can prove the first claim. The second one follows from the Hahn-Banach theorem. The last one is a consequence of the consistency condition (A2). Q.E.D. We further assume the following. (A3) (Normal contraction property) For every f ∈ D, fˇ := 0 ∨ f ∧ 1 belongs to D and kfˇkD ≤ kf kD . (A4) For every f in Db := D ∩ L∞ , f 2 belongs to D. Moreover, sup{kf 2 kD | kf kD + kf kL∞ ≤ 1} is finite. The condition (A4) is equivalent to the following: (A4)’ for every f and g in Db , f g belongs to D. Moreover, sup{kf gkD | kf kD + kf kL∞ ≤ 1, kgkD + kgkL∞ ≤ 1} is finite. Indeed, it is clear that (A4)’ implies (A4). To show the converse implication, use the identity f g = ((f + g)/2)2 − ((f − g)/2)2 and the subadditivity of the norm k · kD . We introduce a sufficient condition for (A3) and (A4). Lemma 2.2. Under (A1) and (A2), the following condition (C) implies (A3) and (A4): (C) when χ is a bounded and infinitely differentiable function on R with χ(0) = 0 and kχ0 k∞ ≤ c for some c > 0, then χ ◦ v ∈ D and kχ ◦ vkD ≤ ckvkD for every v ∈ D. Proof. To show (A3), apply Lemma 2.1 with a sequence {χn ◦v}n∈N so that kχn k∞ ≤ 1 and kχ0n k∞ ≤ 1 for every n, and χn converges pointwise to χ(x) = 0 ∨ x ∧ 1. (A4) is similarly proved. Q.E.D. For f ∈ L0 (X), we set f+ (z) = f (z) ∨ 0 and f− (z) = −(f (z) ∧ 0). Lemma 2.3. Let f ∈ D. Then f+ ∈ D and kf+ kD ≤ kf kD . Proof. Define fn := 0 ∨ f ∧ n = n(0 ∨ (f /n) ∧ 1), n ∈ N. Then kfn kD ≤ nkf /nkD = kf kD by (A3) and fn → f+ pointwise. Lemma 2.1 finishes the proof. Q.E.D. Lemma 2.4. For every f ∈ D, (−a) ∨ f ∧ a → 0 weakly in D as a ↓ 0 and (−a) ∨ f ∧ a → f weakly in D as a → ∞. Proof. It is enough to notice that k(−a) ∨ f ∧ akD = k0 ∨ f ∧ a − 0 ∨ (−f ) ∧ akD ≤ 2kf kD , (−a) ∨ f ∧ a → 0 pointwise as a ↓ 0, (−a) ∨ f ∧ a → f pointwise as a → ∞

4

and to use Lemma 2.1.

M. Hino

Q.E.D.

For a measurable set A, we let DA := {f ∈ D | f = 0 λ-a.e. on X \A} and Db,A := DA ∩ L∞ . A sequence {FS k }k∈N of increasing sets in X is ∞ called a nest if each Fk is closed and k=1 DFk is dense in D. A nest {Fk }k∈N is called (λ-)regular if, for all k, any open set O with λ(O∩Fk ) = 0 satisfies O ⊂ X \ Fk . A subset T∞N of X is called exceptional if there is a nest {Fk }k∈N such that N ⊂ k=1 (X \Fk ). When A is a subset of X, we say that a statement depending on z ∈ A holds quasi everywhere (q.e. in abbreviation) if it does for every z ∈ A \ N for a certain exceptional set N . For a nest {Fk }k∈N , we denote by C({Fk }) the set of all functions f on X such that f is continuous on each Fk . A function f on X is said to be quasi-continuous if there is a nest {Fk }k∈N such that f ∈ C({Fk }). We say that a Borel measure µ on X is smooth if it does not charge any exceptional Borel sets and there exists a nest {Fk }k∈N such that µ(Fk ) < ∞ for all k. A set function ν on X which is given by ν = ν1 −ν2 for some smooth measures ν1 and ν2 with finite total mass is called a finite signed smooth measure. A signed smooth measure ν with attached nest {Fk }k∈N is a map from R := {A ⊂ X | A is a Borel set of some Fk } to R such that ν is represented as ν(A) = ν1 (A) − ν2 (A), A ∈ R, for some smooth Borel measure ν1 and ν2 satisfying νi (Fk ) < ∞ for each i = 1, 2 and k ∈ N. When we want to emphasize the dependency of D, we write D-nest, D-smooth, and so on. We further assume the following quasi-regularity conditions. (QR1) There exists a nest consisting of compact sets. (QR2) There exists a dense subset of D whose elements have quasicontinuous λ-modifications. (QR3) There exists a countable subset {ϕn }n∈N in D and an exceptional set N such that each ϕn has a quasi-continuous λmodification ϕ˜n and {ϕ˜n }n∈N separates the points of X \ N . Every quasi-regular symmetric Dirichlet form (E, F) satisfies all conditions (A1)–(A4) and (QR1)–(QR3) (and (C)) when letting D = F and kf kD = (E(f, f ) + kf k2L2 )1/2 . We give other examples in the last part of this section. Lemma 2.5. There exist some ρ ∈ D and some countable subset C = {hn }n∈N of Db such that 0 ≤ ρ ≤ 1 λ-a.e., hn ≥ 0 λ-a.e. for all n, ˆ | h, h ˆ ∈ C} is dense in D, and for each n ∈ N, ρ ≥ cn hn C − C := {h − h λ-a.e. for some cn ∈ (0, ∞). Proof. Let {fm }m∈N be a countable dense subset of D. Denote by C = {hn }n∈N the set of all arithmetic means of finite number of functions in {(fm )+ ∧ M, (fm )− ∧ M }m∈N, M ∈N . Then C − C is dense in D by

Integral representation

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Lemma 2.4 and the Banach-Saks property. Define ρ=

∞ X

hn . n (kh k + kh k ∞ + 1) 2 n D n L n=1

Then ρ and C satisfy the conditions in the claim.

Q.E.D.

We will fix ρ and C satisfying the statement in the lemma above. Note that we can always take ρ ≡ 1 if 1 ∈ D. Take a strictly increasing and right-continuous function ξ : [0, ∞) → [0, ∞) with ξ(0) = 0. For an open set O ⊂ X, we define (1)

capξ (O) = inf{ξ(kf kD ) | f ∈ D and f ≥ ρ λ-a.e. on O}.

For any subset A of X, we define the capacity of A by capξ (A) = inf{capξ (O) | O ⊃ A, O : open}. It should be noted that capξ (A) ≤ ξ(kρkD ) < ∞ for every A ⊂ X. The following lemma is proved in the same way as in [10]. Lemma 2.6. For every open set O, there exists a unique function eO in D attaining the infimum in (1). Moreover, 0 ≤ eO ≤ 1 λ-a.e. Proof. The uniqueness follows from the uniform convexity of D. The existence is deduced by the Banach-Saks property and (A2). The last claim is a consequence of (A3). Q.E.D. We will discuss some basic properties of the capacity. Lemma 2.7. Let {On }n∈N be a sequence of open sets such that capξ (On ) → 0. Then, there exists a sequence {nk } ↑ ∞ such that eOnk → 0 λ-a.e. Proof. Since keOn kD → 0, the claim is clear from (A2).

Q.E.D.

Lemma 2.8. If capξ (A) = 0, then A ∩ {ρ > 0} is a λ-null set. Proof. Take a decreasing open setsT{On }n∈N such that A ⊂ On and ∞ capξ (On ) → 0. Since eOn ≥ ρ λ-a.e. on k=1 Ok , we have ρ = 0 λ-a.e. on T∞ Q.E.D. k=1 Ok by virtue of Lemma 2.7. This implies the assertion. Lemma 2.9. Let {Ak }k∈N be a sequence of increasing closed sets. Then the following are equivalent. (i) {Ak }k∈N is a nest. (ii) limk→∞ capξ (X \ Ak ) = 0.

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Proof. Suppose (i) holds. Take a sequence {fk }k∈N in D such that fk ∈ DAk and fk → ρ in D. Since ρ − fk = ρ λ-a.e. on X \ Ak , we have capξ (X \ Ak ) ≤ ξ(kρ − fk kD ) → 0 as k → ∞. Next, suppose (ii) holds. It suffices S to prove that each h ∈ C can be approximated in D by functions in k DAk . Since ρ ≥ ch λ-a.e. for some c > 0, it holds that eX\Ak ≥ ch λ-a.e. on X \ Ak for each k. Let fk = (h − c−1 eX\Ak )+ ∈ DAk . By Lemma 2.7, there exists a sequence {k 0 } diverging to infinity such that eX\Ak0 → 0 λ-a.e. Therefore, fk0 → h λ-a.e. as k 0 → ∞. On the other hand, kfk kD ≤ khkD + c−1 keX\Ak kD , which is bounded in k. From Lemma 2.1, we can take S arithmetic means of some subsequence of {fk0 }, which belong to k DAk , so that they converge to h in D. Q.E.D. As is seen from this lemma, any choices of C, ρ and ξ are consistent with the notion of nest. From now on, we treat only the case ξ(t) = t and write cap in place of capξ . Lemma any sequence of subsets {Ak }k∈N in X, it follows ¡S∞ 2.10.¢ For P∞ that cap k=1 Ak ≤ k=1 cap(Ak ). Proof. , . . . , Ok are open sets, it is easy to see the inequal¡Sk When ¢ O1P Pk k ity cap j=1 Oj ≤ j=1 cap(Oj ). Indeed, since j=1 eOj ≥ ρ λ-a.e. Sk on j=1 Oj , we have ° µ[ ¶ °X k k k X X ° ° k ° ≤ ke k = cap(Oj ). Oj ≤ ° eOj ° cap O D j ° j=1

j=1

D

j=1

j=1

Now, let ε > 0. Take an open set Ok for each k ∈ N such that Ok ⊃ Ak Sk and cap(Ok ) < cap(Ak ) + ε2−k . Let Uk = j=1 Oj . Since keUk kD ≤ kρkD < ∞, Lemma 2.1 assures the existence of a subsequence {eUk0 } of {eUk } and e ∈ D such that eUk0 converges to e weakly in D and the arithmetic means of {eUk0 } converge to e in D. Since e ≥ ρ λ-a.e. on S∞ O by using (A2), we have k=1 k µ[ ¶ µ[ ¶ ∞ ∞ Ak ≤ cap Ok ≤ kekD ≤ lim inf keUk0 kD cap 0 k=1

k →∞

k=1

=

lim cap(Uk ) ≤ lim

k→∞

k→∞

k X j=1

Since ε is arbitrary, we obtain the claim.

cap(Oj ) ≤ ε +

∞ X

cap(Ak ).

k=1

Q.E.D.

The following series of lemmas are now proved in a standard way as in the case of quasi-regular Dirichlet spaces; see e.g. [11, 15] for the proof.

Integral representation

7

Lemma 2.11. Suppose that f ∈ D has a quasi-continuous λ-modification f˜. Then, we have cap({f˜ > λ}) ≤ λ−1 kf kD for each λ > 0. Lemma 2.12. (i) When {fn }n∈N is a sequence of quasi-continuous functions, there exists a nest {Fk }k∈N such that fn ∈ C({Fk }) for every n. (ii) If fn ∈ D has a quasi-continuous λ-modification f˜n and converges to f in D as n → ∞, then f has a quasi-continuous λ-modification f˜ and there exists a sequence {nl } ↑ ∞ and a nest {Fk }k∈N such that every f˜n belongs to C({Fk }) and f˜nl converges to f˜ uniformly on each Fk . In particular, f˜nl converges to f˜ q.e. (iii) Every f ∈ D has a quasi-continuous λ-modification f˜. Lemma 2.13. There exists a regular nest {Kk }k∈N such that Kk is a separable and metrizable compact space with respect to the relative topology for any k. Lemma 2.14. Suppose that {Fk }k∈N is a regular nest and S f ∈ C({Fk }). If f ≥ 0 λ-a.e. on an open set O, then f ≥ 0 on O ∩ k Fk . Lemma 2.15. If u1 and u2 are quasi-continuous functions and u1 = u2 λ-a.e., then u1 = u2 q.e. In what follows, f˜ always means a quasi-continuous λ-modification of a function f , a particular version of which is sometimes chosen to suit the context. We can also prove the next two propositions as in [10] (see also [21, Section 2]) by using Lemma 3.1 below together, though they are not used later in this article. Proposition 2.16. For any subset A of X, there exists a unique element eA in the set {f ∈ D | f˜ ≥ ρ˜ q.e. on A} minimizing the norm kf kD . Moreover, 0 ≤ eA ≤ 1 λ-a.e. and cap(A) = keA kD . Proposition 2.17. cap is a Choquet capacity. We remark that the assumption (A4) is not necessary so far. The following are our main theorems, which are stated in [7] in the case of quasi-regular Dirichlet spaces. Theorem 2.18. Under (A1)–(A4) and (QR1)–(QR3), for a bounded linear functional T on D, the next two conditions are equivalent. (i) There exist a nest {Fk }k∈N and positive constants {Ck }k∈N such that for each k ∈ N, |T (v)| ≤ Ck kvkL∞ (X)

for all v ∈ Db,Fk .

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(ii)

There exists a signed smooth measure ν with some attached nest {Fk0 }k∈N such that Z T (v) =

v˜(z) ν(dz) X

for all v ∈

∞ [

Db,Fk0 .

k=1

Moreover, the measure ν is uniquely determined. Theorem 2.19. Under (A1)–(A4) and (QR1)–(QR3), for a bounded linear functional T on D and a positive constant C, the next two conditions are equivalent. (i) |T (v)| ≤ CkvkL∞ (X) for all v ∈ Db . (ii) There exists a finite signed smooth measure ν on X such that the total variation of ν is dominated by C and Z T (v) = v˜(z) ν(dz) for all v ∈ Db . X

In addition, ν is uniquely determined. Moreover, if (C) in Lemma 2.2 holds, we may replace Db in (i) by L that satisfies the following: (L) L is a D-dense subspace of Db such that, for each ε > 0, there is a C ∞ function χ on R with |χ| ≤ 1+ε, 0 ≤ χ0 ≤ 1, χ(x) = x on [−1, 1], and χ ◦ v ∈ L for every v ∈ L. Before ending this section, we give a few examples of D other than quasi-regular Dirichlet spaces. Suppose that X is a separable Banach space and H a separable Hilbert space which is continuously and densely imbedded to X. The inner product and the norm of H will be denoted by h·, ·iH and k · kH , respectively. The topological dual X ∗ of X is identified with a subspace of H. Let λ be a finite Borel measure on X. When K is a separable Hilbert space, we denote by Lp (X → K) the Lp space consisting of K-valued functions on the measure space (X, λ). Define function spaces FCb1 and (FCb1 )X ∗ on X by  ¯  ¯ u(z) = f (`1 (z), . . . , `m (z)),   ¯ FCb1 = u : X → R ¯¯ `1 , . . . , `m ∈ X ∗ , f ∈ Cb1 (Rm ) ,   ¯ for some m ∈ N   ¯ P ¯ G(z) = m gj (z)`j ,   j=1 ¯ (FCb1 )X ∗ = G : X → X ∗ ¯¯ g1 , . . . , gm ∈ FCb1 , ,  ¯ `1 , . . . , `m ∈ X ∗ for some m ∈ N  where Cb1 (Rm ) is the set of all bounded functions f on Rm that have bounded and continuous first-order derivatives. Let u ∈ FCb1 and ` ∈ X ∗ ⊂ H ⊂ X. We define ∂` u by ∂` u(z) = limε→0 (u(z + ε`) − u(z))/ε.

Integral representation

9

The H-derivative ∇u is a unique map from X to H that satisfies the relation h∇u(z), `iH = ∂` u(z), ` ∈ X ∗ ⊂ H. We assume that, if u ∈ FCb1 and v ∈ FCb1 coincide on a measurable set A, then ∇u = ∇v λ-a.e. on A. Let p ≥ 1. We also assume that (∇, FCb1 ) is closable as a map from Lp to Lp (X → H). We denote by W 1,p the domain of the closure of (∇, FCb1 ) and extend the domain of ∇ to W 1,p naturally. The space W 1,p is a separable Banach space with norm kf kW 1,p = kf kLp + k∇f kLp (X→H) . Proposition 2.20. Suppose also p > 1. Then, when we regard W 1,p as D, the conditions (A1)–(A4), (QR1)–(QR3), and (C) are satisfied. Proof. From the results of [21], (A1) and (QR1) hold. Since X is separable, X ∗ is also separable with respect to the weak* topology (see Corollary after Proposition 8 in Chapter IV of [5] for the proof). When {`n }n∈N is a countable dense set of X ∗ , ϕn (·) = arctan `n (·) and N = ∅ assure the validity of (QR3). The remaining conditions are easily checked. Q.E.D. In order to give another example, we introduce real interpolation spaces. Let B0 and B1 be separable Banach spaces. We assume that B0 is continuously imbedded to B1 for simplicity. For parameters q ∈ (1, ∞) and θ ∈ (0, 1), we define the space (B0 , B1 )θ,q by all elements f ∈ B1 such that there exist some Bj -valued measurable functions uj (t) on [0, ∞) (j = 0, 1) satisfying Z ∞ dt (2) u0 (t) + u1 (t) = f a.e. t, < ∞ (j = 0, 1). (tj−θ kuj (t)kBj )q t 0 We set the norm of f ∈ (B0 , B1 )θ,q by · kf k(B0 ,B1 )θ,q = inf

µZ max

u0 , u1 j=0, 1

0

∞

(tj−θ kuj (t)kBj )q

dt t

¶1/q ¸ ,

where the infimum is taken over all pairs u0 and u1 satisfying (2). From the general theory of real interpolation, (B0 , B1 )θ,q is a Banach space, we have continuous imbeddings B0 ,→ (B0 , B1 )θ,q ,→ B1 , and B0 is dense in (B0 , B1 )θ,q . Keeping the notation in the previous example, we have the following proposition. Proposition 2.21. Let p ∈ (1, ∞), q ∈ (1, ∞), and θ ∈ (0, 1). Then D := (W 1,p , Lp )θ,q satisfies (A1)–(A4), (QR1)–(QR3) and (C).

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Proof. In general, we can prove that (B0 , B1 )θ,q is uniformly convex if B0 or B1 is, in the same way as Proposition V.1 of [3]. Therefore, D is uniformly convex. The separability, (QR1) and (QR2) come from those of W 1,p . (QR3) is proved in the same way as the case of W 1,p . (A2) is clearly true. We will prove (C). Let χ be as in (C) in Lemma 2.2. Let f ∈ D and take u0 and u1 satisfying (2). Set v0 (t) = χ ◦ u0 (t) and v1 (t) = χ ◦ f − χ ◦ u0 (t). Then v0 (t) + v1 (t) = χ ◦ f and it is easy to see that kv0 (t)kW 1,p ≤ cku0 (t)kW 1,p and kv1 (t)kLp ≤ cku1 (t)kLp . This implies that χ ◦ f ∈ D and kχ ◦ f kD ≤ ckf kD . Q.E.D. §3.

Proof of Theorems 2.18 and 2.19

First, we will prove that (ii) implies (i) in Theorem 2.18. We take Fk = Fk0 and Ck = |ν|(Fk ) < ∞. Let v ∈ Db,Fk and M = kvkL∞ (X) . We can take a quasi-continuous λ-modification v˜ so that |˜ v | ≤ M everywhere. Then |T (v)| ≤ M |ν|(Fk ) = Ck M . Therefore, (i) holds. Next, we will prove that (i) implies (ii) in Theorem 2.18. Take a (1) (1) (2) (1) nest {Ek }k∈N so that ρ˜ ∈ C({Ek }). Define Ek = Ek ∩ {˜ ρ ≥ 1/k}. (2)

Lemma 3.1. {Ek }k∈N is a nest. (2)

Proof. Clearly, {Ek }k∈N is a sequence of increasing closed sets. Define ρk = ρ ∧ (1/k), k ∈ N. Then ρk → 0 weakly in D by Lemma 2.4. Pm Take a sequence {kj } ↑ ∞ so that ρˆm := (1/m) j=1 ρkj converges to (2)

0 in D as m → ∞. Since ρˆm + eX\E (1) ≥ ρ λ-a.e. on X \ Ekm , we km

(2)

have cap(X \ Ekm ) ≤ kˆ ρm kD + keX\E (1) kD → 0 as m → ∞. Therefore, km

(2)

{Ek }k∈N is a nest. (3)

Q.E.D. (2)

Define Ek = Fk ∩ Kk ∩ Ek , k ∈ N, where Kk is what appeared in (3) Lemma 2.13. Then {Ek }k∈N is a regular nest consisting of separable and metrizable compact sets. Given k ∈ N, let {Uk,n }n∈N be a countable (3) open basis of Ek . The totality of every union of finite elements in {Uk,n }n∈N will be denoted by {Vk,n }n∈N . Take a countable family O of open sets in X such that for every k ∈ N, n ∈ N and ε > 0, some O ∈ O satisfies that O ⊃ Vk,n and cap(O) < cap(Vk,n ) + ε. Recall the condition (QR3) and set S = {(−M ) ∨ ϕ˜n ∧ M | n ∈ N, M ∈ N}. The functions of S separate the points of X \ N . Fix a countable subset D of D such that {f ∈ D | kf kL∞ (X) ≤ M } is a dense set of {f ∈ D | kf kL∞ (X) ≤ M } for each M ∈ N. Take a nest {Fˆk }k∈N so T (3) that Fˆk ⊂ Ek for each k, N ⊂ k∈N (X \ Fˆk ) and the quasi-continuous

Integral representation

11

λ-modifications of all elements in S ∪ D ∪ {eO | O ∈ O} belong to C({Fˆk }). Denote by A the algebra generated by S ∪ {1 ∧ M ρ˜ | M ∈ N}. Note that all functions in A and ρ˜ belong to C({Fˆk }). From the StoneWeierstrass theorem, {f |Fˆk | f ∈ A} is dense in C(Fˆk ) with uniform topology for any k. 0 S Lemma 3.2. There exist a nest {Fk }k∈N and functions {ψn }n∈N in k DFˆk satisfying the following: (i) Fk0 ⊂ Fˆk for all k; (ii) the quasi-continuous λ-modification ψ˜n belongs to C({Fk0 }) for all n; (iii) 0 ≤ ψn ≤ 1 λ-a.e. on X and ψ˜n = 1 on Fn0 for all n. S Proof. Take a sequence {ηn }n∈N ⊂ k DFˆk such that kηn − ρkD < 1/(n2n+1 ), n ∈ N. By Lemma 2.11, there exists an open set Gn so that Gn ⊃ {|˜ ηn − ρ˜| > 1/(2n)} and cap(Gn ) < 2−n for each n. Take a nest {Ek }k∈N such that Ek ⊂ Fˆk and {˜ ηn }n∈N ⊂ C({Ek }). S∞Then η˜n ≥ 1/(2n) on En \Gn since ρ˜ ≥ 1/n on En . Define Fk0 = Ek \ n=k Gn S and ψn = 0 ∨ 2nηn ∧ 1. Then ψn ∈ k DFˆk , ψ˜n = 1 on Fn0 , {Fk0 }k∈N is a sequence of increasing closed P∞ sets, and by Lemma 2.10, we have cap(X \ Fk0 ) ≤ cap(X \ Ek ) + n=k cap(Gn ) → 0 as k → ∞. Q.E.D.

Now, fix n ∈ N and take m ∈ N so that ψn ∈ DFˆm . Define Tn : Db → R by Tn (f ) = T (ψn f ). Since ψn f ∈ Db,Fˆm ⊂ Db,Fm , the statement (i) of Theorem 2.18 implies |Tn (f )| ≤ Cm kψn f kL∞ (X) ≤ Cm kf kL∞ (X) . For an arbitrary f ∈ C(Fˆm ), we can take {fj }j∈N ⊂ A such that limj→∞ kfj − f kC(Fˆm ) = 0. Then, |Tn (fi ) − Tn (fj )| ≤ Cm kψn (fi − fj )kL∞ (X) ≤ Cm kfi − fj kL∞ (Fˆm ) → 0 as i ≥ j → ∞. The limit of {Tn (fj )}j∈N , denoted by Tˆn (f ), satisfies |Tˆn (f )| ≤ Cm kf kC(Fˆm ) . Therefore, Tˆn is a bounded linear functional on C(Fˆm ). On account of the Riesz representation theorem, thereRexists an associated finite signed measure νn on Fˆm such that Tˆn (f ) = ˆ f dνn Fm

for every f ∈ C(Fˆm ). We extend νn to a measure on X by letting νn (A) := νn (A ∩ Fˆm ). Lemma 3.3. The measure νn charges no exceptional sets. Proof. Since the measure |νn | restricted on Fˆm is regular, it is enough to prove that |νn |(K) = 0 for any compact set K ⊂ Fˆm of null capacity. Take such K. Then we can take a sequence {Oj }j∈N from O

12

M. Hino

so that K ⊂ Oj for all j and limj→∞ cap(Oj ) = 0. Indeed, for each j, take an open set O such that K ⊂ O and cap(O) < 1/j. Since K is compact, there is a set V in {Vk,m }k∈N such that K ⊂ V ⊂ O. Choose Oj ∈ O so that V ⊂ Oj and cap(Oj ) ≤ cap(V ) + 1/j. Let fj := eOj . Since {Fˆk } is a regular nest, Lemma 2.14 implies that f˜j = 1 on K ⊂ Oj ∩ Fˆm . We may also assume that 0 ≤ f˜j ≤ 1 everywhere. Since limj→∞ kfj kD = 0, we can suppose fj → 0 λ-a.e. as j → ∞ by taking a subsequence if necessary. Since {f˜j }j∈N is bounded in L2 (|νn |), the arithmetic means {fˆj }j∈N of a further subsequence of {f˜j }j∈N converge strongly in L2 (|νn |). Take a sequence {jl } ↑ ∞ such that fˆjl converges |νn |-a.e. as l → ∞. Define f (z) = lim inf l→∞ fˆjl (z). Then 0 ≤ f ≤ 1 on X, f = 1 on K, and f = 0 λ-a.e. by the way of construction. Given h ∈ A, we have Z (3) fˆjl h dνn = Tˆn (fˆjl h|Fˆm ) = Tn (fˆjl h) = T (ψn fˆjl h). Fˆm

When l tends to ∞, the left-hand side of (3) converges to

R

f h dνn by the dominated convergence theorem. On the other hand, {ψn fˆjl h}l∈N is bounded in D by (A4)’. Since they converge to 0 λ-a.e., they also converge weakly to 0 in D by Lemma 2.1. Therefore, the right-hand R side of (3) converges to 0 as l → ∞. Namely, Fˆm f h dνn = 0. Since {h| ˆ | h ∈ A} is dense in C(Fˆm ), we conclude that f dνn = 0, therefore, Fm

|νn |(K) = 0.

Fˆm

Q.E.D.

Lemma 3.4. For all f ∈ Db , Tn (f ) =

R X

f˜ dνn .

Proof. We can take a sequence {fj }j∈N from D so that {fj }j∈N is bounded in L∞ (X), fj converges to f in D and f˜j converges to f˜ outside some Borel exceptional set N0 . Note that f˜j |Fˆm ∈ C(Fˆm ). Then, Tn (fj ) → Tn (f ) as j → ∞, while Z Z ˜ ˜ ˆ fj dνn = f˜j dνn Tn (fj ) = Tn (fj |Fˆm ) = X X\N0 Z Z j→∞ ˜ −→ f dνn = f˜ dνn X\N0

X

by means of the dominated convergence theorem.

Q.E.D.

For any k, l ∈ N, we have ψ˜k dνl = ψ˜l dνk . Indeed, For f ∈ A, Z Z ˜ f ψk dνl = Tl (f ψk ) = T (ψl f ψk ) = Tk (f ψl ) = f ψ˜l dνk . X

X

Integral representation

13

Therefore, we can define a signed smooth measure ν by ν = νn on Fn0 (n = 1, 2, . . .), which is well-defined by the fact that ψ˜n = 1 on Fn0 . Then for any f ∈ Db,Fk0 , we have ψk f = f and Z Z ˜ T (f ) = T (ψk f ) = Tk (f ) = f dνk = f˜ dν. X

X

Thus, (ii) holds. of ν, it is enough to show that ν ≡ 0 R In order to prove the uniqueness S if X v˜ dν = 0 for all v ∈ k Db,Fk , where {Fk }k∈N is a nest attached with ν. Following the same procedure as in the proof of (i)⇒(ii), take the nests {Fˆk }k∈N and {Fk0 }k∈N , the function space A, and the sequence of functions {ψn }n∈N . For any n ∈ N and f ∈ A, we have f ψn ∈ Db,Fˆn ⊂ R Db,Fn , therefore X f ψ˜n dν = 0. Since {f |Fˆn | f ∈ A} is dense in C(Fˆn ), we have ψ˜n dν = 0. In particular, ν = 0 on Fn0 because ψ˜n = 1 on Fn0 . This implies that ν ≡ 0. The implication (ii)⇒(i) of Theorem 2.19 is proved in the same way as in Theorem 2.18. Because of the result and the proof of (i)⇒(ii) of Theorem 2.18, Theorem 2.19 (i) implies that there exists a finite signed smooth measure ν with some attached nestR{Fk0 }k∈N such that the total S variation is dominated by C and T (v) = X v˜(z) ν(dz) for all v ∈ k Db,Fk0 . It is easy to show that this identity holds for all v ∈ Db by an approximation argument. The uniqueness of ν is clear from the corresponding result of Theorem 2.18. The final claim is also deduced by an approximation argument and the use of Lemma 2.1. This completes the proof of Theorems 2.18 and 2.19. §4.

Application to BV functions on Wiener space

First, we will review some results of [9]. Let E be a separable Banach space and H a separable Hilbert space which is continuously and densely imbedded to E. We use the notations in the end of Section 2 with letting X = E. Define a Gaussian measure µ on E by the following identity: Z √ ` ∈ E ∗ ⊂ H. exp( −1 `(z)) µ(dz) = exp(−k`k2H /2), E

When Y is a separable Hilbert space and ρ is a nonnegative measurable function on E, we denote by Lp (E → Y ; ρ) in this section the Lp space consisting of Y -valued functions on E with underlying measure ρ dµ. We omit E → Y and ρ from the notation when Y = R and ρ ≡ 1, respectively, and write simply Lp for Lp (E → R; 1). We also set

14

M. Hino

T L∞− = p>1 Lp and denote by Lp+ the set of all nonnegative functions in Lp . If u ∈ FCb1 and v ∈ FCb1 coincide on a measurable set A, then ∇u = ∇v µ-a.e. on A. See Proposition I.7.1.4 of [4] for the proof. For p ≥ 1, Clp (E) denotes the set of all functions ρ in L1+ such that (∇, FCb1 ) is closable as a map from Lp (ρ) to Lp (E → H; ρ). A simple example for such ρ is a function which is uniformly away from 0. Suppose ρ ∈ Clp (E). We write W 1,p (ρ) instead of W 1,p when regarding (E, ρ dµ) as (X, λ) in Section 2. When p > 1, W 1,p (ρ) satisfies all the conditions (A1)–(A4), (QR1)–(QR3) and (C). Let F ρ be the topological support of the measure ρ dµ. Since L0 (E → Y ; ρ) is identified with L0 (F ρ → Y ; ρ), we abuse the notation and W 1,p (ρ) is also regarded as a function space on F ρ . When ρ ∈ Cl2 (E), an associated Dirichlet form (E ρ , W 1,2 (ρ)) on L2 (F ρ ; ρ) is defined by Z E ρ (f, g) = h∇f, ∇giH ρ dµ, f, g ∈ W 1,2 (ρ). Fρ

This is a quasi-regular Dirichlet form and a finite signed measure ν on F ρ is smooth with respect to E ρ if and only if ν is W 1,2 (ρ)-smooth. For each G ∈ (FCb1 )E ∗ , the (formal) adjoint ∇∗ G is defined by the following identity: Z Z (∇∗ G)u dµ = hG, ∇uiH dµ for all u ∈ FCb1 . E

E

1/2

Denote by L(log L) the space of all functions f on E such that Φ ◦ |f | ∈ L1 , where Φ(x) = x((log x) ∨ 0)1/2 . We say that a real measurable function ρ on E is of bounded variation (ρ ∈ BV (E)) if ρ ∈ L(log L)1/2 and Z V (ρ) := sup (∇∗ G)ρ dµ < ∞, G

E

where G is taken over all functions in (FCb1 )E ∗ such that kG(z)kH ≤ 1 for every z ∈ E. Let {Tt }t>0 be the Ornstein-Uhlenbeck semigroup, which is associated with E 1 . It is strongly continuous, analytic and contractive on Lp for any p ∈ (1, ∞). We recall some results discussed in [9]. Theorem 4.1. (i) For ρ ∈ BV (E), k∇Tt ρkL1 ≤ V (ρ) for every t > 0. (ii) BV (E) is a vector lattice. Namely, it is a vector space, and for each ρ ∈ BV (E), ρ+ also belongs to BV (E).

Integral representation

(iii)

(iv)

15

A function ρ belongs to BV (E) if and only if ρ ∈ L1 and there exists a sequence {ρn }n∈N in W 1,1 (:= W 1,1 (1)) such that kρn kW 1,1 is bounded in n and ρn → ρ in L1 as n → ∞. Each ρ ∈ BV (E) has a unique finite Borel measure ν and a unique H-valued Borel function σ on E such that kσkH = 1 ν-a.e. and for every G ∈ (FCb1 )E ∗ , Z Z (∇∗ G)ρ dµ = hG, σiH dν. E

E 1,2

The measure ν is W (|ρ|+1)-smooth. If moreover ν ∈ Cl2 (E), then ν|E\F ρ = 0 and ν is W 1,2 (ρ)-smooth. In what follows, we will write νρ for ν in the theorem above. In this section, we improve the result for the smoothness of νρ . In view of the proof of Theorem 4.1 (iv) (Theorem 3.9 of [9]), the smoothness of νρ is derived from the smoothness of ν` for each ` ∈ E ∗ , where ν` is a unique finite signed measure on E satisfying Z Z ∂` u(z)ρ(z) µ(dz) = −2 u(z) ν` (dz), u ∈ FCb1 . E

E

Therefore, applying Theorem 2.19 with L = FCb1 , if we show that the functional Z 1 (4) I` : FCb 3 u 7→ ∂` u(z)ρ(z) µ(dz) ∈ R E

extends continuously on D, where D satisfies (A1)–(A4), (QR1)–(QR3), and (C), and has FCb1 as a dense set, then we can say that ν` , hence νρ , is D-smooth. It is obvious that I` extends to a continuous functional on W 1,p (|ρ|+1) (and W 1,p (ρ) if furthermore ρ ∈ Clp ) for every p ≥ 1. Also, if ρ ∈ Lq for some q ∈ (1, ∞), then I` extends to a continuous functional on W 1,q/(q−1) (:= W 1,q/(q−1) (1)) by H¨older’s inequality. Therefore, we have the following results. Proposition 4.2. Let ρ ∈ BV (E). Then, νρ is W 1,p (|ρ|+1)-smooth for every p > 1. If moreover ρ ∈ Clp , then ν is W 1,p (ρ)-smooth. Proposition 4.3. Let ρ ∈ BV (E) ∩ Lq for some q ∈ (1, ∞). Then, νρ is W 1,q/(q−1) -smooth. In Proposition 4.2, the smaller p is, the stronger the claim is. Now, we will give other examples of D so that νρ is D-smooth. Let us recall the Sobolev spaces in the context of Malliavin calculus. We give several notations in somewhat informal way. We refer to [12] for

16

M. Hino

precise definitions. Let L = −∇∗ ∇ be the Ornstein-Uhlenbeck operator, which is regarded as a generator of {Tt }t>0 . The Sobolev space Dα,p , α ∈ R, 1 < p < ∞, is given by Dα,p = (1 − L)−α/2 (Lp ). Each Dα,p is a separable Banach space with norm kf kDα,p := k(1 − L)α/2 f kLp . The topological dual of Dα,p is identified with D−α,q , q = p/(p − 1). When n ∈ N, by Meyer’s equivalence, ∇n is defined as a continuous operator from Dn,p to Lp (E → H ⊗n ) and k · kLp + k∇n · kLp (E→H ⊗n ) gives a norm on Dn,p which is equivalent to k · kDn,p . In particular, W 1,p (:= W 1,p (1)) is identical with D1,p as a set and their norms are mutually equivalent. We define another Sobolev space Eα,p , α ∈ R, 1 < p < ∞, firstly introduced in [24], by ½ α,p D if α ∈ Z, Eα,p = (Dk+1,p , Dk,p )k+1−α,p if k < α < k + 1, k ∈ Z. The general theory of real interpolation implies that (Eα,p )∗ is identified with E−α,q , where q = p/(p − 1) (see also [24]). When 0 < α < 1 and 1 < p < ∞, Eα,p satisfies conditions (A1)–(A4), (QR1)–(QR3), and (C) by virtue of Proposition 2.21, if Eα,p is equipped with a norm deduced by (W 1,p , Lp )1−α,p . For such indices, FCb1 is dense in Eα,p since W 1,p is dense in Eα,p . For later use, following [1, 2], we introduce another equivalent norm on Eα,p based on the K-method by µZ 1 ¶1/p dε , kf k0Eα,p = (ε−α K(ε, f ))p ε 0 where © ª K(ε, f ) = inf kf1 kLp + εkf2 kW 1,p | f = f1 + f2 , f1 ∈ Lp , f2 ∈ D1,p . The connection between BV (E) and Eα,p is given as follows. Theorem 4.4. Let q > 1. Then BV (E) ∩ Lq ⊂ Eα,p if 1 < p < q and α < (1/p − 1/q)/(1 − 1/q). Also, this inclusion is continuous when BV (E) ∩ Lq is equipped with norm kf kBV (E)∩Lq = V (f ) + kf kLq . In particular, BV (E) ∩ L∞− ⊂ Eα,p if p > 1 and αp < 1. For the proof, we need the following estimates. Lemma 4.5. (i) When θ/a + (1 − θ)/b = 1/p with 0 < θ < 1, a, b, p ≥ 1, we have kf kLp ≤ kf kθLa kf k1−θ . Lb (ii) For each r ≥ 0 and p ∈ (1, ∞), there exists some C such that k(1 − L)r Tt f kLp ≤ Ct−r kf kLp for every t ∈ (0, 1] and f ∈ Lp . Proof. The claim (i) follows from a simple application of H¨older’s inequality. The claim (ii) is a consequence of Theorem 6.13 (c) of Chapter 2 in [18], since {Tt }t>0 is an analytic semigroup on Lp . Q.E.D.

Integral representation

17

Proof of Theorem 4.4. Let f ∈ BV (E)∩Lq with V (f )+kf kLq ≤ 1. In the following, ci denotes a constant depending only on p and q. By Theorem 4.1 (i), k∇Tt f kL1 ≤ V (f ) ≤ 1 for any t > 0. By virtue of Meyer’s equivalence and Lemma 4.5 (ii), for t ∈ (0, 1], k∇Tt f kLq ≤ c1 k(1 − L)1/2 Tt f kLq ≤ c2 t−1/2 . Applying Lemma 4.5 (i) with a = 1 and b = q, that is, θ = (1/p − 1/q)/(1−1/q), we have k∇Tt f kLp ≤ (c2 t−1/2 )1−θ for t ∈ (0, 1], therefore, kTt f kW 1,p ≤ c3 t−(1−θ)/2 .

(5) From the identity

Z f − Tt f

t

d Ts f ds = − ds

Z

t

LTs f ds = − 0 0 Z t {((1 − L)1/2 Ts/2 )2 f − Ts f } ds, = 0

we obtain, for t ∈ (0, 1], Z (6)

kf − Tt f kLp

t

≤ 0

Z

t

≤ 0

Z

t

≤ 0

Z

t

≤

k((1 − L)1/2 Ts/2 )2 f kLp ds + tkf kLp c4 s−1/2 k(1 − L)1/2 Ts/2 f kLp ds + t c5 s−1/2 kTs/2 f kW 1,p ds + t c6 s−1/2 s−(1−θ)/2 ds + t ≤ c7 tθ/2 .

0

Here we used Lemma 4.5 (ii) in the second line and (5) in the last line. By combining (5) and (6), for each ε ∈ (0, 1], K(ε, f ) ≤ kf − Tε2 f kLp + εkTε2 f kW 1,p ≤ c8 εθ , and, if α ∈ (0, θ), µZ

1

(ε

−α

p dε

K(ε, f ))

0

This proves the claim.

ε

¶1/p ≤ c8 {p(θ − α)}−1/p < ∞. Q.E.D.

Using Theorem 4.4, we obtain the Eα,p -smoothness of νρ by the following proposition.

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M. Hino

Proposition 4.6. Let ρ ∈ BV (E) ∩ Lq , q > 1. Then the map I` in (4) extends continuously on Eα,p if p > q/(q − 1) and αp > q/(q − 1). Therefore, νρ is Eα,p -smooth for such α and p with α ∈ (0, 1). In particular, if ρ ∈ BV (E) ∩ L∞− , then νρ is Eα,p -smooth for any α, p with α ∈ (0, 1) and αp > 1. Proof. Due to Meyer’s equivalence, the map u 7→ ∂` u is continuous from D1,p to Lp and from Lp to D−1,p , respectively. By the real interpolation theorem, it is continuous from Eα,p to Eα−1,p for any α ∈ (0, 1). The claim follows from the fact (E1−α,p/(p−1) )∗ = Eα−1,p and BV (E) ∩ Lq ⊂ E1−α,p/(p−1) by the assumption and Theorem 4.4. Q.E.D. Remark 4.7. (i) In [24], it is proved that Dα+ε,p ,→ Eα,p ,→ α−ε,p for every α ∈ R, 1 < p < ∞ and ε > 0. Therefore, D Theorem 4.4 and Proposition 4.6 remain valid if we replace Eα,p by Dα,p . (ii) When ρ ∈ BV (E) is an indicator function of some set A, νρ can be regarded as a surface measure of A. The smoothness of νρ that is proved in the proposition above is consistent with Theorem 9 of [6] saying that the Hausdorff measure of codimension n on Wiener space does not charge any set of (α, p)-capacity as long as p > 1 and αp > n. Lastly, we give a few nontrivial examples of BV functions, referring to the work [2]. Note that by combining Theorem 4.8 and Theorem 4.4 we recover a part of the results in [2]. Theorem 4.8. (i) Let F be a function such that F ∈ D2,p −1 and k∇F kH ∈ Lq for some p > 1 and q > 1 with 1/p+1/q < 1. Let A = {F < x} with x ∈ R. Then 1A ∈ BV (E). (ii) Suppose that (E, H, µ) is a classical Wiener space on [0, 1]. For x > 0, set A = {w ∈ E | max0≤s≤1 |w(s)| < x}. Then 1A ∈ BV (E). Proof. (i): From the assumptions, we have ∇∗ (∇F/k∇F kH ) ∈ La for some a > 1. Indeed, keeping in mind the fact k∇F kH , k∇2 F kH⊗H ∈ Lp due to Meyer’s equivalence, let ε tend to 0 in the identity à ! ∇F LF h∇F ⊗ ∇F, ∇2 F iH⊗H ∗ p ∇ . = −p + 2 2 (k∇F k2H + ε)3/2 k∇F kH + ε k∇F kH + ε R∞ Now, set ψn (y) = n1[x−1/n,x] (y), ϕn (y) = y ψn (z) dz, and ρn = ϕn (F ). Then we have Z k∇ρn kL1 (E→H) = ψn (F )k∇F kH dµ E

Integral representation

19

À Z ¿ ∇F ∇ρn , dµ k∇F kH H E ¶ µ Z ∇F dµ = − ρn ∇ ∗ k∇F kH E ° µ ¶° ° ∗ ° ∇F ° , ≤ ° ∇ ° k∇F kH °L1 = −

which is bounded in n. Since {ρn }n∈N is uniformly bounded and converges to 1A pointwise, Theorem 4.1 (iii) completes the proof. (ii): Set ρn (w) = 0 ∨ n (1 − max0≤s≤1 |w(s)|/x) ∧ 1, w ∈ E for each n ∈ N. By the calculation in the proof of Theorem 3.1 of [2], we have ρn ∈ W 1,1 and kρn kW 1,1 is bounded in n. Since {ρn }n∈N is uniformly bounded and tends to 1A pointwise, Theorem 4.1 (iii) completes the proof. Q.E.D.

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