Normal trace for vector fields of bounded mean oscillation

TL;DR

This paper introduces new $BMO$ spaces for vector fields with bounded mean oscillation, analyzing their boundary traces and divergence properties, revealing differences between normal and tangential components.

Contribution

It defines novel $BMO$ vector field spaces with bounded normal trace and divergence control, highlighting distinct behaviors of normal and tangential components.

Findings

Normal trace of $BMO$ vector fields is bounded under divergence control.

Zero extension of the normal component remains in $BMO$.

Tangential components may not share the same extension property.

Abstract

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.