Let's define the harmonic oscillator $H = -\Delta+x^2$ in a domain $\Omega$ of $\mathbb R^d$. Thus, we consider the Dirichlet eigenvalue problem $$ (H - \lambda^2)u (x) = 0, \ x \in \Omega ; \ \text{ and } \ u(x) = 0 \ x \in \partial \Omega.$$
Denote the spectral projection to the eigenvalue $ k = \lambda^2 $ by $P_k$.

My question is: Are there any $L^p$ bounds results for eigenfunctions of the Hermite operator $H$ that have the form
$$ \| P_k u \|_{L^p(\Omega)} \leq k^{\rho(p)} \| u\|_{L^2(\Omega)},$$ where $p$ would be in $[2, 2d/(d-2)]$ and the exponent $\rho$ as function of $1/p$.

Note that in the case of $\Omega = \mathbb R^d$ there are many results, Karadzhov 1994, Thangavelu 1998, Koch and Tataru 2004.