$\newcommand{\R}{\mathbb{R}}$ $\DeclareMathOperator{\law}{Law}$
$\def\E{\hskip.15ex\mathsf{E}\hskip.10ex}$ Let $H$ be a Hilbert space equipped with the orthonormal basis $(e_i)_{i\ge1}$. Let $\xi=\sum_{k=1}^\infty \sigma_ke_k\xi_k$ be a Gaussian random variable on this space; here $\sigma_i\in\R$, and $\xi_i$ are i.i.d standard Gaussian random variables with mean $0$ and variance $1$, $\sum \sigma_i^2<\infty$.
It is not very difficult to show that for any bounded measurable function $f\colon H\to H$ one has \begin{equation}\tag{1} \|\E f(x+\xi)-\E f(\xi)\|_H\le d_{TV}(\law(\xi),\law(\xi+x))\le C \|f\|_{\mathcal{C}^0(H,H)}\|x\|_{CM},\quad x\in H, \end{equation} where $\|x\|_{CM}$ is the Cameron-Martin norm of $x$, that is $$ \|x\|^2_{CM}:=\sum_{i=1}^\infty \sigma_i^{-2}x_i^2. $$
It is also immediate to show that for any Lipschitz function $f\colon H\to H$ one has \begin{equation}\tag{2} \|\E f(x+\xi)-\E f(\xi)\|_H\le C \|f\|_{\mathcal{C}^1(H,H)}\|x\|_{H},\quad x\in H. \end{equation}
Question. I would like to derive a similar inequality for functions $H\to H$ that are intermediate between bounded and Lipschitz. More precisely, for $f\in C^\gamma(H,H)$ with $\gamma\in(0,1)$, I would like to obtain an estimate of the form $$ \|\E f(x+\xi)-\E f(\xi)\|_H\stackrel{???}{\le} C \|f\|_{\mathcal{C}^\gamma(H,H)}\|x\|_{H}^\gamma\|x\|_{CM}^{1-\gamma},\quad x\in H. $$
However, it is not at all clear how to prove this. In particular, this does not follow automatically from (1) and (2) since interpolation inequalities fail in infinite dimensions. Although they hold in finite dimensions, their constants typically blow up as the dimension of the space tends to infinity.
