On a characterization of the Sobolev spaces over an abstract Wiener space
Let us recall the classical Sobolev spaces in a finite-dimensional case. Consider \(H^ 1({\mathbb{R}}^ d)\) for example. Usually it is defined by means of the Schwartz distribution, that is, (A) \(H^ 1({\mathbb{R}}^ d)\equiv \{f\in L^ 2({\mathbb{R}}^ d)\); For each \(i=1,...,d\), the distribution derivative \(\frac{\partial}{\partial x_ i}f\) belongs to \(L^ 2({\mathbb{R}}^ d)\}.\) \(H^ 1({\mathbb{R}}^ d)\) is a Hilbert space with norm \(\| f\|_{H^ 1}\equiv (\| f\|^ 2_{L^ 2}+\sum^{d}_{i=1}\| \frac{\partial}{\partial x_ i}f\|^ 2_{L^ 2})^{1/2}.\) However, if we have to define it without the notion of distribution, we may adopt the following definition. (B) \(H^ 1({\mathbb{R}}^ d)\equiv\) the completion of the space \(C^ 1_ 0({\mathbb{R}}^ d)\) with respect to \(\|.\|_{H^ 1}.\) Or, due to Nikodym, we can take the next one. (C) \(H^ 1({\mathbb{R}}^ d)\in \{f\in L^ 2({\mathbb{R}}^ d)\); For each \(i=1,...,d\), there exists a version \(\tilde f_ i\) of \(f\) such that \(\tilde f_ i\) is absolutely continuous along almost all lines parallel to the \(x_ i\)-axis, and its Radon-Nikodym derivative \(\frac{\partial}{\partial x_ i}\tilde f_ i\) belongs to \(L^ 2({\mathbb{R}}^ d).\}\) Now, talking about the Sobolev spaces over an abstract Wiener space, two typical definitions are known; one is due to \textit{J. Shigekawa} [this J. 20, 263-289 (1980; Zbl 0476.28008)], and the other is due to \textit{S. Kusuoka} and \textit{D. Stroock} [North-Holland Math. Libr. 32, 271-306 (1984; Zbl 0546.60056)]. In short words, we can say that the former definition is an infinite-dimensional analogue of type (B), and the latter one is that of type (C). In this paper, we first present a theorem in which Shigekawa's Sobolev spaces are characterized, in an analogous way to (A), by means of so-called generalized Wiener functionals. Then, as its application, we will prove that those two definitions of Shigekawa and Kusuoka-Stroock in fact determine the same spaces.
Sobolev spaces over an abstract Wiener space
scientific article; zbMATH DE number 3994286