Characterization of the traces on the boundary of functions in magnetic Sobolev spaces

Abstract We characterize the trace of magnetic Sobolev spaces defined in a half-space or in a smooth bounded domain in which the magnetic field A is differentiable and its exterior derivative corresponding to the magnetic field dA is bounded. In particular, we prove that, for d ≥ 1 and p > 1 , the trace of the magnetic Sobolev space W A 1 , p ( R + d + 1 ) is exactly W A ∥ 1 − 1 / p , p ( R d ) where A ∥ ( x ) = ( A 1 , … , A d ) ( x , 0 ) for x ∈ R d with the convention A = ( A 1 , … , A d + 1 ) when A ∈ C 1 ( R + d + 1 ‾ , R d + 1 ) . We also characterize fractional magnetic Sobolev spaces as interpolation spaces and give extension theorems from a half-space to the entire space.