Solution
= Solution
The homogeneous <Sobolev embedding theorem> gives
$$
\dot H^\sigma(\mathbb R^d)\hookrightarrow L^{p+1}(\mathbb R^d),
\qquad
\frac1{p+1}=\frac12-\frac\sigma d,
$$
so
$$
\sigma=\frac d2-\frac d{p+1}
=\frac{d(p-1)}{2(p+1)}.
$$
Now $s_c>0$ is equivalent to $p-1>4/d$, which gives $\sigma>2/(p+1)$. Similarly $s_c<1$ is equivalent to $(d-2)(p-1)<4$, which gives $\sigma<1$.