Let $f \in (L^2 (\mathbb R^n))^n$, i.e. $f$ is a vector-valued function. We assume $\nabla \cdot f = 0$ and $\int (1+\mid x \mid)\mid f(x) \mid dx < \infty$.
How can I show that $$f \in (\dot B_2^{-\frac{n+2}{2},\infty})^n \iff \sup_{j \in \mathbb Z}\int_{\frac{1}{2} \leq 2^j \mid \xi \mid \leq 1} \mid \mathcal F(f) \mid^2 \frac{d\xi}{\mid \xi \mid^{n+2}} < \infty.$$
where $\mathcal F$ denotes the Fourier transform.
I just started learning Besov spaces, and looked up relevant notes, but still can't show the equivalence above.
Updated.