On subordination conditions for systems of minimal differential operators

In this paper, we provide a review of results on a priori estimates for systems of minimal differential operators in the scale of spaces $L^p(\Omega),$ where $p\in[1,\infty].$ We present results on the characterization of elliptic and $l$-quasielliptic systems using a priori estimates in isotropic and anisotropic Sobolev spaces $W_{p,0}^l(\mathbb R^n),$ $p\in[1,\infty].$ For a given set $l=(l_1,\dots,l_n)\in\mathbb N^n$ we prove criteria for the existence of $l$-quasielliptic and weakly coercive systems and indicate wide classes of weakly coercive in $W_{p,0}^l(\mathbb R^n),$ $p\in[1,\infty],$ nonelliptic, and nonquasielliptic systems. In addition, we describe linear spaces of operators that are subordinate in the $L^\infty(\mathbb R^n)$-norm to the tensor product of two elliptic differential polynomials.