For $s\in[0,1]$ define function spaces
$H^s(\mathbb{R})=\{u\in L_2(\mathbb{R}): (1+|\cdot|^2)^{s/2}\mathcal{F}u\in L_2(\mathbb{R}) \}$ (where $\mathcal{F}$ denotes the Fourier transform) i.e. the space of Bessel potentials.
$E^s(\mathbb{R})=\{u\in H^s(\mathbb{R}): |\cdot|^{-s}u\in L_2(\mathbb{R})\}$
Prove that
- $E^s(\mathbb{R})=H^s(\mathbb{R})$ for $s\in[0,1/2)$
- $E^s(\mathbb{R})=H^s_0(\mathbb{R}):=\{u\in H^s(\mathbb{R}): u(0)=0\}$ for $s\in(1/2,1]$ (notice that $u(0)$ is well defined since $H^s(\mathbb{R})\subset C(\mathbb{R})$ for $s>1/2$).
Thank You for any comments.
It is not a homework. For the case $s\in[0,1/2)$ I have tried to use Sobolev embedding $H^s\subset L_q$ where $q=2/(1-2s)$ and Holder inequality. Then one gets $\int|x|^{-2s}|u(x)|^2\leq(\int|u(x)|^{2q/2})^{2/q}(\int|x|^{-2s(q/2)'})^{(2/q)'}$, where $1=1/(q/2)+1/(q/2)'$. Unfortunately $(q/2)'=(q/2)/(q/2-1)=q/(q-2)=1/(2s)$ so that on the right hand side we get $\int|x|^{-1}=\infty$.