The Helmholtz Decomposition of a BMO Type Vector Field in a Slightly Perturbed Half Space

Abstract

We introduce a space of $$L^2$$ L 2 vector fields with bounded mean oscillation whose “normal” component to the boundary is well-controlled. We establish its Helmholtz decomposition in the case when the domain is a perturbed $$C^3$$ C 3 half space in $${\textbf{R}}^n$$ R n $$(n \ge 3)$$ ( n ≥ 3 ) with small perturbation.