I'm looking for a reference about the Gagliardo-Nirenberg-Sobolev interpolation inequality on a half space $\mathbb{R}^d_+:= \lbrace x \in \mathbb{R}^d \mid x_d > 0 \rbrace$ :
Let $1 \leq p,q,r \leq \infty$ and $m \in \mathbb{N}$. Suppose $j \in \mathbb{N}$ and $\alpha \in [0,1]$ satisfy the relations \begin{align*} &\dfrac{1}{p}=\dfrac{j}{d}+\left( \dfrac{1}{r}-\dfrac{m}{d} \right)\alpha+\dfrac{1-\alpha}{q},\\ &\dfrac{j}{m} \leq \alpha \leq 1, \end{align*} with the exception $\alpha<1$ if $m-j-d/r \in \mathbb{N}$.
Then for all $u \in \mathrm{L}^q(\mathbb{R}^d_+)$, if $\mathrm{D}^m u \in \mathrm{L}^r(\mathbb{R}^d_+)$ and if $u$ has a vanishing trace on $\partial\mathbb{R}^d_+$, then we have $$ \Vert \mathrm{D}^j u \Vert_{\mathrm{L}^p(\mathbb{R}^d_+)} \lesssim \Vert \mathrm{D}^m u \Vert_{\mathrm{L}^r(\mathbb{R}^d_+)} ^{\alpha} \Vert u \Vert_{\mathrm{L}^q(\mathbb{R}^d_+)}^{1-\alpha},$$ where $\lesssim$ is a universal constant.
I know that this inequality holds in the case of $\mathbb{R}^d$ or a smooth bounded domain (with a function having a vanishing trace on the boundary) but I was wondering if it also holds for a half space.