Consider the Gaussian Sobolev space $W^{p,r}(\gamma_n)$. It is the completion of $C_c^\infty(\mathbb{R}^n)$ with respect to the norm $$ \|f\|_{s,p}=\sum_{|\alpha|=0}^r\|\partial^\alpha f\|_p, $$ where $\|\cdot\|_p$ is the $p$-norm with respect to the standard Gaussian measure $\gamma_n$ on $\mathbb{R}^n$. Equivalently, it consists of all $f\in L^2(\gamma_n)$ such that the distributional derivatives up to order $s$ are regular and in $L^p(\gamma_n)$.
Does the operator $T\colon f\mapsto xf$ map $W^{1,2}(\gamma_1)$ or even $L^2(\gamma_1)$ into $L^2(\gamma_1)$?
If so, is it continuous as an operator from the respective domain to $L^2(\gamma_1)$?