On the normal trace space of extended divergence-measure fields

We characterise the normal trace space associated to extended (measure-valued) divergence-measure fields on the boundary of a set $E\subset {\mathbb{R}}^n$, as the Arens-Eells space $\mathrm{A}\mathrm{E}(\mathrm{\partial }E)$. Such a trace operator is constructed for any Borel set $E$, and under a mild regularity condition, which includes Lipschitz domains, this trace operator is shown to moreover be surjective. This relies in part on a new pointwise description of the Anzellotti pairing $\overline{\mathrm{\nabla }\phi \cdot \mathit{F}}$ between a ${\mathrm{W}}^{1,\mathrm{\infty }}$ function $\phi$ and extended divergence-measure field $\mathit{F}$. As an application, we prove extension theorems for divergence-measure fields and divergence-free measures. Results for ${\mathrm{L}}^1$-fields are also obtained.