On subordination conditions for systems of minimal di erential 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 $L^p(\mathrm{\Omega }),$ where $p\in [1,\mathrm{\infty }].$ We present results on the characterization of elliptic and $l$-quasielliptic systems using a priori estimates in isotropic and anisotropic Sobolev spaces $W_{p,0}^l({\mathbb{R}}^n),$ $p\in [1,\mathrm{\infty }].$ 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 $W_{p,0}^l({\mathbb{R}}^n),$ $p\in [1,\mathrm{\infty }],$ nonelliptic, and nonquasielliptic systems. In addition, we describe linear spaces of operators that are subordinate in the $L^{\mathrm{\infty }}({\mathbb{R}}^n)$-norm to the tensor product of two elliptic differential polynomials.