The Helmholtz decomposition of a $BMO$ type vector field in general unbounded domains

Abstract

We consider a space of $L^2$ vector fields with bounded mean oscillation whose ``normal'' component to the boundary is well-controlled. In the case when the dimension $n\ge 3$, we establish its Helmholtz decomposition for arbitrary uniformly $C^3$ domain in ${\mathbf{R}}^n$.

41 pages, 3 figures