论文标题
在球体上的局部支持的矢量场的硬性组件之间的关系
Relation between Hardy components for locally supported vector fields on the sphere
论文作者
论文摘要
鉴于在球体上的内谐梯度的耐用空间中，我们考虑了在球体外部谐波梯度的耐用空间中找到相应函数的问题，H-（S）使得两个函数的总和与本地支持的矢量场的总和仅通过切向偏差差异而不同。我们表征了H+（s）的子空间，该子空间允许这种延续并表明它是密度但在H+（s）内不关闭的子空间。此外，我们得出线性映射，该线性映射将矢量字段从H+（S）子空间映射到H-（S）中的相应唯一矢量场。显式结构使用层电位，但涉及无界操作员。我们指出了一些有限的极端问题，这些问题支持对耐力组件之间的映射进行数值评估。研究此问题的最初动机来自与本地化约束的反磁化问题。
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.
