For which $t$ and $p$ does function of the form $$ f(x) = 1/|x|^{\alpha} ,\quad 0<\alpha<1$$ belong to the fractional Sobolev space $H_p^t(\mathbb{R})$, $p>1,t\geq 0$?
UPD: actually I am intersted in $$ f(x) = \chi_{[-1,1]}/|x|^{\alpha} ,\quad 0<\alpha<1$$ where $\chi_{[-1,1]}$ is a characteristic function of the interval $[-1,1]$.