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

We characterize the trace of magnetic Sobolev spaces defined in a half-space or in a smooth bounded domain in which the magnetic field Ais differentiable and its exterior derivative corresponding to the magnetic field dAis bounded. In particular, we prove that, for d >= 1and p > 1, the trace of the magnetic Sobolev space W-A(1, p)(R-+(d+1)) is exactly W-A parallel to(1-1/p,p) (R-d) where A(parallel to) (x) =(A(1),..., A(d))( x, 0) for x is an element of R-d with the convention A =(A(1),..., A(d+1)) when A is an element of C-1(R-+(d+1), Rd+1). We also characterize fractional magnetic Sobolev spaces as interpolation spaces and give extension theorems from a halfspace to the entire space. (C) 2020 Elsevier Inc. All rights reserved