Hardy-Hodge decomposition of vector fields in $\mathbb {R}^n$

Abstract:

We prove that an $\mathbb {R}^{n+1}$-valued vector field on $\mathbb {R}^n$ is the sum of the traces of two harmonic gradients, one in each component of $\mathbb {R}^{n+1}\setminus \mathbb {R}^n$, and of an $\mathbb {R}^n$-valued divergence free vector field. We apply this to the description of vanishing potentials in divergence form. The results are stated in terms of Clifford Hardy spaces, the structure of which is important for our study.