Relation between Hardy components for locally supported vector fields on the sphere

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.