Normal trace for vector fields of bounded mean oscillation

We introduce various spaces of vector fields of bounded mean oscillation ($BMO$) defined in a domain so that normal trace on the boundary is bounded when its divergence is well controlled. The behavior of “normal” component and “tangential” component may be different for our $BMO$ vector fields. As a result zero extension of the normal component stays in $BMO$ although such property may not hold for tangential components.