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

TL;DR

This paper investigates the relationship between Hardy components of vector fields on the sphere, characterizing when a continuation exists, and provides explicit constructions and potential numerical methods, motivated by inverse magnetization problems.

Contribution

It characterizes the subspace of Hardy space functions allowing continuation between inner and outer harmonic gradients on the sphere and constructs the associated linear mapping explicitly.

Findings

The subspace allowing continuation is dense but not closed in H+(S).

An explicit layer potential-based construction of the mapping is provided.

Potential numerical evaluation methods for the mapping are discussed.

Abstract

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.