  Given a function in the Hardy space of inner harmonic gradients on the
sphere, H+(S), we consider the problem of finding a corresponding function in
the Hardy space of outer harmonic gradients on the sphere, H-(S), such that the
sum of both functions differs from a locally supported vector field only by a
tangential divergence-free contribution. We characterize the subspace of H+(S)
that allows such a continuation and show that it is dense but not closed within
H+(S). Furthermore, we derive the linear mapping that maps a vector field from
this subspace of H+(S) to the corresponding unique vector field in H-(S). The
explicit construction uses layer potentials but involves unbounded operators.
We indicate some bounded extremal problems supporting a possible numerical
evaluation of this mapping between the Hardy components. The original
motivation to study this problem comes from an inverse magnetization problem
with localization constraints.
