I know (and it is straightforward to show) that the magnetic Sobolev norm
$$\|f\|_{H^2_A(\mathbb{R}^n)} := \sqrt{ \|f\|_{L^2(\mathbb{R}^n)}^2 + \|(-i\nabla +A)^2f\|_{L^2(\mathbb{R}^n)}^2} $$
(works also for $H^1_A(\mathbb{R}^n)$) is equivalent to the usual $H^2(\mathbb{R}^n)$ ($H^1(\mathbb{R}^n)$ respectively) norm if $A$ is bounded on $\mathbb{R}^n$, i.e $A\in L^{\infty}(\mathbb{R}^n)$ and $A\in C^1(\mathbb{R}^n)$. My question is: Are there weaker conditions for $A$ other than being bounded (and continuously differentiable) such that the statement still holds?
To add detail why it is true for $A$ being bounded: You basically use the following fact: Let $A\in L^{\infty}(\mathbb{R}^3)$. Then for all $f\in H^2_A(\mathbb{R}^3)$ $$\|f\|_{H^2_A(\mathbb{R}^3)} = \|f\|_{L^2(\mathbb{R}^3)} + \|(-i\nabla+A)^2f\|_{L^2(\mathbb{R}^3)} \leq \|f\|_{L^2(\mathbb{R}^3)} + \|\Delta f\|_{L^2(\mathbb{R}^3)} + \|2(A\cdot \nabla)f\|_{L^2(\mathbb{R}^3)} + \|A^2f\|_{L^2(\mathbb{R}^3)} \leq C_1\|f\|_{L^2(\mathbb{R}^3)} +C_2\|\nabla f\|_{L^2(\mathbb{R}^3)} +\|\Delta f\|_{L^2(\mathbb{R}^3)}$$
which leads to the claim. This strategy obviously works if and only if $A$ is bounded. But is there maybe any other way to go without assuming that $A$ is bounded?