# Trace theorem for magnetic Sobolev spaces

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.

• The space depends heavily on the behavior of $A$ at $\infty$. With no assumption on $A$ you cannot expect a precise answer concerning $H_A^1$. The curvature of $A$ may also introduce more subtle constraints. – Liviu Nicolaescu Jan 30 '16 at 12:55
• @LiviuNicolaescu Of course my $A$ should be divergent at infinity, since a constant $B=\operatorname{curl} A$ must be allowed. Do you have any reference? – Siminore Jan 30 '16 at 14:52
• The answer depends on the behavior of $A$ at $\infty$. – Liviu Nicolaescu Jan 30 '16 at 17:59
• For instance $A(x_1,x_2,x_3)=(-x_2,x_1,0)$. – Siminore Jan 30 '16 at 19:15

## 1 Answer

If the magnetic field $$\operatorname{curl} A$$ is bounded, then the trace space is $$\left\{ g \in L^2 (\mathbb{R}^{N - 1}, \mathbb{C}) ; \iint\limits_{\mathbb{R}^{N - 1} \times \mathbb{R}^{N - 1}} \frac{\lvert e^{-i \int_{0}^1 A ((1 -t)x + t y)) \,\mathrm{d} t \cdot (y - x)} g (y) - g (x)\rvert^2}{\lvert y - x \rvert^N} \,\mathrm{d}y \, \mathrm{d} y < +\infty \right\};$$ with estimates depending only on $$\lVert \operatorname{curl} A \rVert_{L^\infty}$$. See our paper Nguyen, Van Schaftingen, Characterization of the traces on the boundary of functions in magnetic Sobolev spaces, arXiv:1905.01188.