For integers $n \ge 1$ and $m \ge 0$, the Sobolev space $W^{m,2}(\mathbb R^n)$ is characterized by $$ f \in W^{m,2}(\mathbb R^n) \text{ iff } \tilde f_m \in L^2(\mathbb R^n), \label{1}\tag{1} $$
where $\tilde f_m:\mathbb R^n \to \mathbb C$ is defined by $\tilde f_m(z) := (1 + \|z\|^2)^{m/2}|\widehat f(z)|$, and $\widehat f$ is the Fourier transform of $f$.
One way to see this at least in the case $m=1$ is to note that if the $f:\mathbb R^n \to \mathbb R$ is a continuously-differentiable square-integrable function, and we define another function $g : \mathbb R^n \to \mathbb R$ by $g(x):=\|\nabla f(x)\|$, then $\hat g(z) = -i\|z\| \hat f(z)$ for all $z \in \mathbb R^n$, and Parseval's identity gives $$ \begin{split} \|f\|_{L^2(\mathbb R^n)}^2 &= \|\widehat f\|_{L^2(\mathbb R^n)}^2 = \int_{\mathbb R^n} |\widehat f(z)|^2 \mbox{d}z,\\ \|\nabla f\|_{L^2(\mathbb R^n)}^2 &= \|g\|_{L^2(\mathbb R^n)}^2 = \|\widehat g\|_{L^2(\mathbb R^n)}^2 = \int_{\mathbb R^n}\|z\|^2 |\widehat f(z)|^2\mbox{d}z. \end{split} $$
Thus, $$ \begin{split} \|f\|_{W^{1;2}(\mathbb R^n)}^2 &:= \|f\|_{L^2(\mathbb R^n)}^2 + \|\nabla f\|_{L^2(\mathbb R^n)}^2 = \int_{\mathbb R^n}(1+\|z\|^2)|\widehat f(z)|^2\mbox{d} z\\ & = \int_{\mathbb R^n}|\widetilde f_2(z)|^2\mbox{d} z = \|\widetilde f_2\|_{L^2(\mathbb R^n)}^2. \end{split} $$
Question. Let $\gamma_n$ be the standard Gaussian measure on $\mathbb R^n$. Is there a characterization similar to \eqref{1} for the weighted Sobolev space $W^{m,2}(\mathbb R^n, \gamma_n)$ ?
I'm particularly interested in the case $m=1$.
N.B.: It might help to note that $\widehat \gamma_n = \gamma_n$.
Conjecture. The computations here https://mathoverflow.net/a/396157/78539, if correct, would suggest that the sought-for characterization would be something like
$$ f \in W^{1,2}(\mathbb R^n,\gamma_n) \text{ iff } \tilde f_2 \in L^2(\mathbb R^n,\gamma_n). $$
