Consider Sobolev spaces $$ H^s(\mathbb R^d)=\{f\in \mathcal{S}'(\mathbb R^d): \mathcal{F}^{-1} [\langle \cdot \rangle^s \mathcal{F}(f)] \in L^2(\mathbb R^d) \}$$ 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: Let $0<s<1/2.$ Can we say that $$H^s(\mathbb R^d) \subset \bigcap_{2<p<\infty} L^p(\mathbb R^d)$$ ? If not, any counter example?