Characterization of Product Measures by Integrability Condition

Let (E ′, μ) be the real Gaussian space, where E ′ is the space of tempered distributions and μ be the standard Gaussian measure on E ′. In the recent papers [4, 7, 8], Asai, Kubo and Kuo (AKK for short) have shown that in order to construct the Gel’fand triple [E ]u ⊂ L2(E ′, μ) ⊂ [E ]u associated with a growth function u ∈ C+,1/2, essential conditions on u are (U0)(U2)(U3) stated in Section 2.2. Legendre transform and dual Legendre transform (Section 2.1) play important roles to get this result. We note that Gannoun et al. [11] have obtained similar results independently. Some relationships with [11] are discussed in Section 2.2. In addition, the intrinsic topology for [E ]u has been given and the characterization theorem for positive Radon measures on E ′ has also been proved by considering an integrability condition [5, 8]. Now it is natural to ask whether “positivity” of white noise operators can be discussed in some sense and characterized. To answer this question, we consider the Gel’fand triple over the Complex Gaussian space (E ′ c, μc), i.e. E ′ c = E ′ + iE ′ equipped with the product measure μc = μ ′ × μ′ where μ′ is the Gaussian measure on E ′ with variance 1/2 (Section 2.2). Following AKK’s Legendre transform technique, we have Wu1,u2 ⊂ L2(E ′ c, μc) ⊂ [W ]u1,u2 for functions u1, u2 ∈ C+,1/2 satisfying (U0)(U2)(U3). Several examples for u1, u2 are given in Section 2.3. We remark that Ouerdiane [28] studied a special case u1(r ) = u2(r ) = exp(k−1rk), where 1 ≤ k ≤ 2. In Section 3, the characterization theorem for measures can be extended to the case of positive product Radon measures on E ′ × E ′. In addition, the notion of pseudo-positive operators is naturally introduced via kernel theorem and characterized by an integrability condition. Lemma 3.2 plays crucial roles in Section 3.