I am looking for a $C^\infty$ function $g:\mathbb{R}^3\to \mathbb{R}^3$ such that $g(x)=0$ for $|x|\le 1$ and $g(x)=x$ for $|x|\ge 2$. Certainly such $g$ can be constructed, but I also want it to satisfy the additional property that for each $j=1,2,3$,
$$\sum _{k=1}^3 \left |\frac{\partial ^2 g_k}{\partial x_j \partial x_k}\right | \le C\sum _{i=1}^3 \left | \frac{\partial g_i}{\partial x_i}\right |$$
for some constant $C>0$.
Can I find $g$?