The Sard inequality on Wiener space
Let \((W,H,\mu)\) be an abstract Wiener space. For a measurable map \(u:W\to H\), \(T(w):= w+u(w)\), \(w\in W\) is the Cameron-Martin perturbation of the identity map. We denote by \(\mathbb{D}_{p,k}(H)\) the Sobolev space in the Malliavin calculus with \(p>1\), \(k\in \mathbb{N}\). The \(H\)-valued random variable \(u(w)\) is said to be \(\eta-H- C^1\) if there exists a nonnegative random variable \(\eta\) such that \(\mu\{\eta> 0\}>0\) and for all \(w\in Q:=\{w: \eta(w)>0\}\), \(u(w+ h)\) is Fréchet differentiable on the random open ball \(\{h\in H: |h|_H< \eta(w)\}\). For \(F\in \mathbb{D}^{\text{loc}}_{p,1}(H)\), we define \[ \Lambda_F(w):= \text{det}_2(I+\nabla F)\exp(-\delta F-\textstyle{{1\over 2}} |F|^2_H), \] where \(\nabla F\) is Hilbert-Schmidt and \(\text{det}_2\) indicates the Carleman-Fredholm determinant. \(\delta\) denotes the adjoint of the gradient \(\nabla\) under the Wiener measure \(\mu\), and is called the Itô-Ramer integral (or divergence or Skorokhod integral). The authors first present detailed proofs of the measurability of the forward images of Borel sets under the Cameron-Martin perturbation of the identity maps \(T(w)\). Moreover, an infinite-dimensional version of the Sard inequality is shown: Suppose that \(u: W\to H\) is a measurable map in \(\mathbb{D}_{p,1}(H)\) and is \(\eta- H-C^1\). Then for any \(A\in{\mathcal B}(W)\), \(\mu(T(A\cap Q))\leq \int_{A\cap Q}|\Lambda_u|d\mu\). Arguments used in the proof are technically due to the authors [Probab. Theory. Relat. Fields 99, No. 4, 485-500 (1994; Zbl 0801.60031)]. Lastly, applications to absolute continuity on Wiener space as well as to the change of variable formula are presented. Those are closely related to previous results of the authors [ibid. 105, No. 4, 545-563 (1996; Zbl 0853.60036)]. Some results in this paper are applied to degree theory on the Wiener space in a recent paper of the authors [ibid. 108, No. 2, 259-279 (1997; Zbl 0883.60037)].
scientific article; zbMATH DE number 1078498
scientific article