I am considering a smooth vector field $A \colon \mathbb{R}^N \to \mathbb{R}^N$ and I define the magnetic Sobolev space $H_A^1$ as the closure of $C_0^\infty(\mathbb{R}^N)$ with respect to the norm $$ u \mapsto \int_{\mathbb{R}^N} \left| \left( -i\nabla -A \right)u \right|^2 + |u|^2. $$ My question is: suppose that I need to replace $\mathbb{R}^N$ by $\mathbb{R}_{+}^{N+1}=[0,+\infty) \times \mathbb{R}^N$. Does there exist a characterization of the trace space associated to $H_A^1$, if of course I identify $\partial \mathbb{R}_{+}^{N+1}$ with $\mathbb{R}^N$?

The answer should depend somehow on $A$, which I assume to be locally bounded but not necessarily globally bounded.