For reference, I am reading the paper "Uniqueness of Finite Energy Solutions for Maxwell-Dirac and Maxwell-Klein-Gordon Equations" by Masmoudi and Nakanishi.

Let $A\_0$ be a scalar function satisfying the Poisson equation

$$\Delta A\_0 = -|u|^2$$

where $u$ is taken from the space $X^{1/2,b} = U(t)H^b\_t H^{1/2}\_x + U(-t)H^b\_t H^{1/2}\_x$ (formally) where $U(\pm t) = \exp(\pm i|\nabla|t)$ is the free wave propagator. 

The authors claim the following estimate holds true
$$ \|A\_0\|\_{L^{\infty}(\dot{B}^{3/2}\_{2,1})} \lesssim \| |u|^2 \|\_{L^{\infty}(\dot{B}^{-1/2}\_{2,1})} \lesssim \| u\|^2\_{L^{\infty}H^{1/2}} $$

Here, $\dot{B}$ denotes the usual homogeneous Besov space.

I have two questions:

1. Why do we lose two derivatives in the first estimate. Shouldn't we gain two? Or am I just confused here.
2. I completely fail to understand the second estimate. Is there an embedding $H^{1/2} \hookrightarrow \dot{B}^{-1/2}\_{2,1}$? The only thing I know is that $H^m$ is isomorphic to $B^m\_{22}$ for $m\in \mathbb{N}$ but here we have non-integer $s$.

EDIT: As described in the comments under this post, I changed the last term in the chain of inequalities.