Limiting Korn-Maxwell-Sobolev inequalities for general incompatibilities

We give sharp conditions for the limiting Korn-Maxwell-Sobolev inequalities \begin{align*} \lVert P\rVert_{\dot{W}{^{k-1,\frac{n}{n-1}}}($\mathbb{R}$^n)}\le c\big(\lVert\mathscr{A}[P]\rVert_{\dot{W}{^{k-1,\frac{n}{n-1}}}($\mathbb{R}$^n)}+\lVert$\mathbb{B}$P\rVert_{L^{1}($\mathbb{R}$^n)}\big) \end{align*} to hold for all $P\in C_c^{\mathrm{\infty }}({\mathbb{R}}^n;V)$, where $\mathcal{A}$ is a linear map between finite dimensional vector spaces and $\mathbb{B}$ is a $k$-th order, linear and homogeneous constant-coefficient differential operator. By the appearance of the $L^1$-norm of the differential expression $\mathbb{B}P$ on the right-hand side, such inequalities generalise previously known estimates to the borderline case $p=1$, and thereby answer an open problem due to Müller, Neff and the second author (Calc. Var. PDE, 2021) in the affirmative.