On the complex constant rank condition and inequalities for differential operators

In this note, we study the complex constant rank condition for differential operators and its implications for coercive differential inequalities. These are inequalities of the form $$\Vert A u \Vert_{L^p} \leq \Vert \mathscr{A} u \Vert_{L^q},$$ for exponents $1\leq p,q <\infty$ and homogeneous constant-coefficient differential operators $A$ and $\mathscr{A}$. The functions $u \colon \Omega \to \mathbb{R}^d$ are defined on open and bounded sets $\Omega \subset \mathbb{R}^N$ satisfying certain regularity assumptions. Depending on the order of $A$ and $\mathscr{A}$, such an inequality might be viewed as a generalisation of either Korn's or Sobolev's inequality, respectively. In both cases, as we are on bounded domains, we assume that the Fourier symbol of $\mathscr{A}$ satisfies an algebraic condition, the complex constant rank property.