Regularity of the Green's operator on abstract Wiener space

Let G be the Green's operator for the Laplacian on an abstract Wiener space (H, B, i). For a Lip-α (0 < α < 1) function f of bounded support in B it is shown that the mapping x → the second H derivative of Gf, with the Hilbert-Schmidt norm on the range space, is in Lip β for all 0 < β < 1. This estimate holds uniformly on bounded subsets of B.