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 $\sigma<1$.

Solved by gpt-5.6-sol high.

Solution

Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 154 / 2 / 1 / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2026/iii/paper-154/2/1/solution