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I came across this inequality on the Gaussian space recently. I was not aware of its existence since it is not really a classical one in comparison to the Poincaré inequality or the Logarithmic Sobolev inequality but it seems to be useful in order to prove the analyticity of the Ornstein-Uhlenbeck semigroup in $L^p(\gamma)$. Let $\gamma$ be the standard Gaussian measure on $\mathbb{R}^d$. Let $p \in (1,+\infty)$, let $f\in \mathcal{S}(\mathbb{R}^d)$ and let $k \in \{1, \dots, d\}$. Then, \begin{align*} \|x_kf\|_{L^p(\gamma)} \leq C_{p,d} \left(\|f\|_{L^p(\gamma)}+\|\partial_k(f)\|_{L^p(\gamma)}\right), \end{align*} where $C_{p,d}>0$ only depending on $d$ and on $p$.