We define energy space $$E= \left\{ f\in \mathcal{S}'(\mathbb R^d):\|\nabla f\|_{L^2} + \|xf\|_{L^2} < \infty \right\}.$$
and Sobolev spaces $$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^d): \mathcal{F}^{-1} [\langle \cdot \rangle^s \mathcal{F}(f)] \in L^p(\mathbb R^d) \}$$ where $\langle \cdot \rangle = (1+ |\cdot|^2)^{1/2}, s\in \mathbb R,$ and $\mathcal{F}$ and $\mathcal{F}^{-1}$ are Fourier transform and the inverse Fourier transform.
My Question is: Is there any relation between $E$ and $L^p_s (p\neq 2)$? Specifically, I'm interested to know the relation between $E$ and $L^p_s$ for $s>d(\frac{2}{p}-1), 1\leq p <2$?
Side Thoughts: I think there is no any relations between $L^p-$spaces on $\mathbb R^d,$ I'm guessing there might no relations between $E$ and $L^{p}_s (p\neq 2).$ But I do not know how to construct counter examples?
Motivation: This spaces appear in PDE very often.
Rough Ideas in constructing counter example: Define $$f_p(x)= \sum_{k\neq 0} |k|^{-\frac{d}{p}-\epsilon} e^{ik\cdot x} e^{-|x|^2}$$ in $\mathcal{S}'(\mathbb R^d) \ (1< p<2).$
Can we say that $f_p\in E$?
For simplicity we consider one dimension case, that is, $d=1.$
We note that
$$\nabla f(x) =f'(x)= \sum_{k\neq 0} |k|^{-\frac{d}{p}-\epsilon} e^{ik\cdot x} e^{-|x|^2}=\sum_{k\neq 0} |k|^{-\frac{d}{p}-\epsilon} (-2|x|e^{ik\cdot x} e^{-|x|^2} + ik e^{ik\cdot x} e^{-|x|^2} ) $$
Now
$\|\nabla f\|_{L^2}^2 = \int_{\mathbb R} \left|\sum_{k\in \mathbb Z \setminus \{0\} } |k|^{-\frac{d}{p}-\epsilon} (-2|x|e^{ik\cdot x} e^{-|x|^2} + ik e^{ik\cdot x} e^{-|x|^2} ) \right|^2 dx$
I do not know how to proceed but my guess is there might exists $1<p<2$ such that $f_p \notin E.$