Metric differentiability of Lipschitz maps defined on Wiener spaces

This note is devoted to the differentiability properties of H-Lipschitz maps defined on abstract Wiener spaces and with values in metric spaces, so we start by recalling some basic definitions related to the Wiener space structure. Let (E, ‖ · ‖) be a separable Banach space endowed with a Gaussian measure γ. Recall that a Gaussian measure γ on E equipped with its Borel σ−algebra B is a probability measure on (E,B) such that the law (pushforward measure) of each continuous linear functional on E is Gaussian, that is, γ ◦ (e∗)−1 is a Gaussian measure on R for each e∗ ∈ E∗ \ {0}, possibly a Dirac mass. If we assume, as we shall do, that γ is not supported in a proper subspace of E, then all such measures are Gaussian measures. We shall also assume, for the sake of simplicity, that γ is centered, i.e. ∫ E xdγ(x) = 0. The Cameron Martin space associated to (E, γ) can be defined, as a vector space, by