We define $L^p-$ Sobolev spaces as follows: $$L^p_s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^d): \mathcal{F}^{-1} [\langle \cdot \rangle^s \mathcal{F}(f)] \in L^p(\mathbb R^d) \}$$ 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.
When $p=2,$ we put $L^p_s=H^s.$
My Question is: (I) Can we find $s_1>0$ and $2<p<\infty$ and $f\in L^p_{s_1}(\mathbb R^d)$ such that $f\notin H^s(\mathbb R^d)$ (for $0<s<1/2$)? (II)Can we find $s_1>0$ and $2<p<\infty$ and $f\in L^p_{s_1}(\mathbb R^d)\cap L^2(\mathbb R^d)$ such that $f\notin H^s(\mathbb R^d)$ (for $0<s<1/2$)?
My Guess: I think, this is possible to do. As we know thee is no relation between $L^2(\mathbb R^d)$ and $L^p(\mathbb R^d)$ ($p\neq 2$). I do not know how to deal the situation in $L^p-$Sobolev spaces.
