$C^\infty$ extension operator

Question

Let $B_r$ denotes the ball $B(0,r) \subset \mathbb{R}^3$ and $h \in ]0,1]$.

I am looking for the following extension operator :

$$P:C^\infty(B_h) \rightarrow C^\infty(B_2)$$

such that for all $u \in C^\infty(B_h)$, $P(u) \in C^\infty(B_2)$ verifies the following properties, for a fixed $s>3$

• $P(u)=u$ in $B_h$

• $||\nabla P(u)||_{(L^s(B_2))^3} \leq C_1 ||\nabla u||_{L^s(B_h)}$, with $C_1$ a positive constant independant of $h$ and $u$

My guess is that it might comes from a Sobolev operator extension. Any help or thoughts are welcomed !

Answer 1

This cannot work for just $C^\infty$-functions. Take $u(x):=\frac1{h^2-\|x\|^2}$. It is infinitely often differentiable and unbounded on $B_h$. So it cannot be extended to $B_2$.