Consider the Sobolev spaces with $p=2$, defined for $s \in \mathbb{R}$ as \begin{equation} W^{s} = \left\{ u \in \mathcal{S}', \ (1 + \lvert \cdot \rvert^2)^{{s}/{2}} \widehat{u} \in L_2 \right\}. \end{equation} It is a Hilbert space for the norm $\lVert u \rVert_{W^s} = \left(\int_{\mathbb{R}} (1 + \lvert \xi \rvert^2)^s \lvert \widehat{u} (\xi) \rvert^2 \mathrm{d} \xi \right)^{1/2}$.
The weighted Sobolev space $W^{s,r}$ is now defined for $s,r \in \mathbb{R}$ as \begin{equation} W^{s,r} = \left\{ u \in \mathcal{S}', \ (1 + \lvert \cdot \rvert^2)^{{r}/{2}} u \in W^s \right\}. \end{equation} Again, it is a Hilbert space for the norm $\lVert u \rVert_{W^{s,r}} = \lVert (1 + \lvert \cdot \rvert^2)^{{r}/{2}} u \rVert_{W^s}$.
We have the obvious embeddings, for $s_1 \leq s_2$ and $r_1 \leq r_2$, \begin{align} W^{s_2,r} \subseteq W^{s_1,r}, \\ W^{s,r_2} \subseteq W^{s,r_1}. \end{align}
Now, is the following result true?
Conjecture: Fix $s_1, s_2, r_1, r_2 \in \mathbb{R}$. Then, \begin{equation} W^{s_1,r_1} \cap W^{s_2,r_2} = W^{\max(s_1,s_2),\max(r_1,r_2)}. \end{equation}
Of course, due to the embeddings above, $W^{\max(s_1,s_2),\max(r_1,r_2)}$ is included in $W^{s_1,r_1}$ and $W^{s_2,r_2}$ and therefore in their intersection. Is the other inclusion also valid?
