Constant rank operators in Korn-Maxwell-Sobolev inequalities

We focus on Korn-Maxwell-Sobolev inequalities for operators of reduced constant rank. These inequalities take the form $$\mathrm{\parallel }P-{\mathrm{\Pi }}_{\mathbb{B}}{\mathrm{\Pi }}_{\ker \mathcal{A}}P{\mathrm{\parallel }}_{{\dot{\mathrm{W}}}^{k-1,p^{\ast }}({\mathbb{R}}^n)}\le c(\mathrm{\parallel }\mathcal{A}[P]{\mathrm{\parallel }}_{{\dot{\mathrm{W}}}^{k-1,p^{\ast }}({\mathbb{R}}^n)}+\mathrm{\parallel }\mathbb{B}P{\mathrm{\parallel }}_{{\mathrm{L}}^p({\mathbb{R}}^n)})$$ for all $P\in {\mathrm{C}}_c^{\mathrm{\infty }}({\mathbb{R}}^n;V)$, where $V$ is a finite-dimensional vector space, $\mathcal{A}$ is a linear mapping, and $\mathbb{B}$ is a constant coefficient homogeneous differential operator of order $k$. In particular, we can treat the combination $(p,\mathcal{A},\mathbb{B},k)=(1,\mathrm{tr},\mathrm{Curl},1)$. Our results generalize the techniques from Gmeineder et al. (Math.Mod.Met.Appl.Sci,34:03,2024; arXiv:2405.10349), which exclusively dealt with reduced elliptic operators. In contrast to the reduced ellipticity case, however, the reduced constant rank case necessitates to introduce a correction, namely the projection ${\mathrm{\Pi }}_{\mathbb{B}}$ on the left-hand side of the inequality.