Limiting Korn-Maxwell-Sobolev inequalities for general incompatibilities

We give sharp conditions for the limiting Korn-Maxwell-Sobolev inequalities {align*} P_{{Ẇ}{k-1,n/n-1}(R^n)}≤ c([P]_{{Ẇ}{k-1,n/n-1}(R^n)}+_{L¹(R^n)}) {align*} to hold for all P∈ Cc∞(Rⁿ;V), where A is a linear map between finite dimensional vector spaces and B is a k-th order, linear and homogeneous constant-coefficient differential operator. By the appearance of the L¹-norm of the differential expression BP 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 M\"{u}ller, Neff and the second author (Calc. Var. PDE, 2021) in the affirmative.