Let $X=H_r^p(\mathbb{R},\mu)$ be the Sobolev space of order $r$ with respect to the Gaussian measure $\mu$.
Is the operator $d\colon H_1^p(\mathbb{R},\mu)\to L^p(\mathbb{R},\mu)$ given by $df(x)=-f'(x)+xf(x)$ continuous?
I know that for $f,g\in C_c^\infty(\mathbb{R})$ and $p=2$, $d$ is the adjoint of the derivative, i.e.
$$(f',g)=(f,dg),$$
where $(\cdot,\cdot)$ is the $L^2(\mathbb{R},\mu)$ inner product.
Since $X$ is the closure of $C_c^\infty(\mathbb{R})$ in the respective Sobolev norm, I feel that this might be useful for the case $p=2$, although I haven't been able to prove it.
Nonetheless, I am also interested in the case $p\not=2$ and don't see how the duality helps in that case.