About the weighted Sobolev norm

Question

I'm wondering that the Sobolev norm with weight $$ \|f\|_{H^{s,k}(\mathbb R^n)} := \|(1+|x|^2)^{k/2} (\sqrt{1-\Delta})^s f \|_{L^2(\mathbb R^n)} $$ is equivalent to the norm $$ \|(\sqrt{1-\Delta})^s(1+|x|^2)^{k/2} f \|_{L^2(\mathbb R^n)}. $$ Here $f(x):\mathbb R^n \to \mathbb R$ and $\Delta$ is Laplacian, $k,s >0$.

Answer 1

Yes it is. Sketch of proof: 1) induction on nonnegative integer $k,s$ (trivial for $k=0$ and for $s=0$; after that, induction $k\to k+1$ using commutation of $x$ with $\Delta$, and likewise for $s\to s+1$).

2) For negative $k,s$ by duality. (Omit step 2 if you only need nonnegative $k$ and $s$.)

3) Now for arbitrary (not necessarily integer) $k,s$ by interpolation.