Coercive estimates for multilayer degenerate differential operators

We obtain the conditions under which a given multilayer differential operator $P(D)$ (polynomial $P(\xi)$) is more powerful than operator $Q(D)$ (polynomial $Q(\xi)$). This is used to obtain estimates of monomials, which, in turn, using the theory of Fourier multipliers, is used to obtain coercive estimates of derivatives of functions through the differential operator $P(D)$ applied to these functions.