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Integral Representation of Linear Functionals on Vector Lattices and Its Application to BV Functions on Wiener Space

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Integral Representation of Linear Functionals on Vector Lattices and Its Application to BV Functions on Wiener Space 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 rep- resentation 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)| ≤ CkkvkL for all v ∈ Fb,Fk , k ∈ N holds for some nest {Fk}k∈N and some constants Ck, k ∈ N. As its ap- plications, 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, Sci- ence and Technology, Grant-in-Aid for Encouragement of Young Scientists, 15740089, 2003. 2 M. Hino which lie in this framework are first order Sobolev spaces derived from a gradient operator and fractional order Sobolev spaces by a real interpo- lation 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 modifica- tions 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 p p spaces and their norms, such as L (X) (or simply L ) and k · kLp . 0 We suppose that a subspace D of L (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 Banach- Saks 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, kfkD ≤ lim infk→∞ 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 ˇ belongs to D and kfkD ≤ kfkD. ∞ 2 (A4) For every f in Db := D ∩ L , f belongs to D. Moreover, 2 sup{kf kD | kfkD + kfkL∞ ≤ 1} is finite. The condition (A4) is equivalent to the following: (A4)’ for every f and g in Db, fg belongs to D. Moreover, sup{kfgkD | kfkD + kfkL∞ ≤ 1, kgkD + kgkL∞ ≤ 1} is finite. Indeed, it is clear that (A4)’ implies (A4). To show the converse im- plication, use the identity fg = ((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 0 with χ(0) = 0 and kχ 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 0 so that kχnk∞ ≤ 1 and kχnk∞ ≤ 1 for every n, and χn converges pointwise to χ(x) = 0 ∨ x ∧ 1. (A4) is similarly proved. Q.E.D. 0 For f ∈ L (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 ≤ kfkD. Proof. Define fn := 0 ∨ f ∧ n = n(0 ∨ (f/n) ∧ 1), n ∈ N. Then kfnkD ≤ nkf/nkD = kfkD 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 ≤ 2kfkD, (−a) ∨ f ∧ a → 0 pointwise as a ↓ 0, (−a) ∨ f ∧ a → f pointwise as a → ∞ 4 M. Hino and to use Lemma 2.1. 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 {Fk}k∈N of increasing sets in X is S∞ 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 N of X is called exceptional if there is a T∞ 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 quasi- continuous λ-modifications. (QR3) There exists a countable subset {ϕn}n∈N in D and an ex- ceptional set N such that each ϕn has a quasi-continuous λ- modificationϕ ˜n and {ϕ˜n}n∈N separates the points of X \ N. Every quasi-regular symmetric Dirichlet form (E, F) satisfies all con- ditions (A1)–(A4) and (QR1)–(QR3) (and (C)) when letting D = F and 2 1/2 kfkD = (E(f, f) + kfkL2 ) . 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.
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