Inequality involving magnetic derivative I need to understand the following inequality:
\begin{align*}
\|\partial_j \psi\|_2 &\leq \|D_j\psi\|_2 + \|A_j\|_6\|\psi\|_3 \\ &\leq \|D_j\psi\|_2 + C \sum_{k=1}^3\|\partial_k A_j\|_2\sum_{k=1}^3\|\partial_k \psi\|^{1/2}_2\|\psi\|^{1/2}_2
\end{align*}
We are in $\mathbb{R}^3$ and $D_j = \partial_j-iA_j$, $j=1,2,3$. The first inequality follows from triangle and Hölder inequalities but I have a hard time seeing the second one. I tried Sobolev embedding but that gives (for example) $\|A_j\|_6 \leq \|A_j\|_{H^1}$ and not $\|A_j\|_6 \leq \|\nabla A_j\|_2$. For the other terms I tried $$\|\psi\|_3 =\|\psi^{1/2}\psi^{1/2}\|_3 \leq \|\psi^{1/2}\|_{12}\|\psi^{1/2}\|_4 = \|\psi\|^{1/2}_6\|\psi\|_2^{1/2}$$ by Hölder but again I don't know how to establish $\|\psi\|_6\leq \|\nabla\psi\|_2$.
 A: I will just assume that the $A_j$ are smooth enough. Then the Gagliardo-Nirenberg-Sobolev inequality
$$
||u||_{L^{p^*}(\mathbb{R^n})} \leq  C ||Du||_{L^{p}(\mathbb{R^n})}
$$
gives you
$$
||A_j||_{L^{6}(\mathbb{R^n})} \leq C \sum_k ||\partial_k A_j||_{L^{2}(\mathbb{R^n})}
$$
since the exponents match perfectly: Here $p^*$ is the usual Sobolev conjugate
$$
p^*=\frac{np}{n-p}  \iff 6=\frac{2\cdot3}{3-2}
$$
The second part follows from the Hölder interpolation inequality (I found it in my German textbook, shouldn't be too hard to find)
$$
||\psi||_{L^{q}(\mathbb{R^n})} \leq ||\psi||_{L^{p}(\mathbb{R^n})}^{\theta} ||\psi||_{L^{p^*}(\mathbb{R^n})}^{1-\theta}
$$
Plugging in the exponents, we have
$$
\theta=\frac{\frac{1}{q}-\frac{1}{p}}{\frac{1}{p^*}-\frac{1}{p}}=\frac{1/3-1/2}{1/6-1/2}=\frac{1}{2}
$$
And so
$$
||\psi||_{L^{3}(\mathbb{R^n})} \leq ||\psi||_{L^{2}(\mathbb{R^n})}^{\frac{1}{2}} ||\psi||_{L^{6}(\mathbb{R^n})}^{\frac{1}{2}}
$$
Then you acn again use the first inequality to see that
$$
||\psi||_{L^{3}(\mathbb{R^n})} \leq C^{\frac{1}{2}}\sum_k||\psi||_{L^{2}(\mathbb{R^n})}^{\frac{1}{2}} ||\partial_k \psi||_{L^{2}(\mathbb{R^n})}^{\frac{1}{2}}
$$
Combining all the estimates you have your desired estimates.
A note on the inequality:
Gagliardo-Nirenberg-Sobolev inequality only holds for $C_c^{\infty}$; however, you can find a smooth, compactly supported sequence of functions $\psi_i \to \psi$ in $W^{1,p}(\mathbb{R}^n)$. Then you have
$$
||\psi_i -\psi_m||_{L^{p^*}(\mathbb{R^n})} \leq  C ||D\psi_i -D\psi_m||_{L^{p}(\mathbb{R^n})}
$$
and so the sequence also converges in $L^{p^*}(\mathbb{R^n})$, since Cauchy sequences converge. Take limits and conclude.
