Canonical RT0 interpolation on H1 fields

Assume Countable Choice, a conforming affine simplicial mesh ${\mathcal{T}}_h$ from Affine simplicial meshes and RT/BDM spaces, and $v\in H^1(D;{\mathbb{R}}^d)$. The facet trace inequality of H1 facet trace inequality on shape-regular simplices makes each moment $m_F(v):={\int }_F{\gamma }_{F}v\cdot n_F$ finite. On each cell $K$, use its outward normals to prescribe the $d+1$ facet moments (with sign reversed when $n_F$ points inward to $K$); by RT0 flux unisolvence in RT0 facet-flux unisolvence and global H(div) conformity, there is a unique $I_{K}v\in {\mathrm{RT}}_0(K)$ with those moments. Define $I_{h}v$ by $(I_{h}v){\mathrm{\mid }}_K=I_{K}v$. The elementwise traces of the global H1 field agree almost everywhere across each shared facet by H1 traces agree across interior mesh facets, so the shared moment is single-valued in the fixed orientation; the same unisolvence result gives $I_{h}v\in {\mathrm{RT}}_0({\mathcal{T}}_h)\subset H(\mathrm{div};D)$. No canonical facet-moment definition on rough H(div)-only input is made here.