On the Besov space

Question

I want to prove this:

$$ C^{-1}\lambda^{s-\frac dp}\lVert u\rVert\smash{\dot B}_{p,r}^s \,\le \, \lVert u(\lambda\cdot)\rVert_{\smash{\dot B}_{p,r}^s} \,\le\, C\lambda^{s-\frac dp}\lVert u\rVert\smash{\dot B}_{p,r}^s. $$

(original screenshot)

I proved it for $\lambda = 2^k$ for every $k$ integer!

Can you help me to conclude..

Thank You..

Answer 1

You can get the proof on from this lecture (Proposition 1.5.4) .