Uniformly bounded commuting projection for lowest-order Nedelec spaces

Assume Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a nonempty bounded Lipschitz polyhedron and let $\{{\mathcal{T}}_h\}$ be a shape-regular family of matching affine tetrahedral meshes with $h={\max }_{K\in {\mathcal{T}}_h}\mathrm{diam}(K)\to 0$. For the full lowest-order spaces $E_h^0=W_h^0,E_h^1=N_h^1,E_h^2=N_h^2,E_h^3=N_h^3$ of Lowest-order Whitney and Nedelec de Rham spaces, there are linear projections ${\pi }_h^k:L^2{\mathrm{\Lambda }}^k(\mathrm{\Omega })\to E_h^k$ satisfying ${\pi }_h^k{\mathrm{\mid }}_{E_h^k}=I$ and $d{\pi }_h^k={\pi }_h^{k+1}d$ on $H{\mathrm{\Lambda }}^k(\mathrm{\Omega })$ for $k=0,1,2$. In degree three the exterior derivative is zero. Their norms are uniformly bounded both in $L^2$ and in $H{\mathrm{\Lambda }}^k$, for $k=0,1,2,3$.

There is also a boundary-corrected family of projections ${\pi }_{h,0}^k:L^2{\mathrm{\Lambda }}^k(\mathrm{\Omega })\to E_{h,0}^k$, where $E_{h,0}^k=E_h^k\cap H_0{\mathrm{\Lambda }}^k(\mathrm{\Omega })$ and $H_0{\mathrm{\Lambda }}^k$ is the graph closure of compactly supported smooth forms. These have the same uniform bounds, are the identity on $E_{h,0}^k$, and commute in degrees zero through two on the homogeneous complex; the terminal degree-three differential is zero. In particular, ${\pi }_{h,0}^0:H_0^1(\mathrm{\Omega })\to W_h^0\cap H_0^1(\mathrm{\Omega }),{\pi }_{h,0}^1:H_0(\mathrm{curl};\mathrm{\Omega })\to N_h^1\cap H_0(\mathrm{curl};\mathrm{\Omega }),\mathrm{\nabla }{\pi }_{h,0}^0={\pi }_{h,0}^1\mathrm{\nabla }\mathrm{.}$ Here the zero tangential space uses Zero tangential trace H(curl) space. Each family converges strongly in $L^2$ on every $L^2$ form, uniformly on compact $L^2$ subsets, and in the graph norm on its corresponding graph domain. The full and homogeneous families are separate operators. These are cochain projections; canonical edge moments are not asserted bounded on $H(\mathrm{curl})$.