Coercive estimates for multilayer-degenerate differential operators (polynomials)

Let $P(D) = \sum_{\alpha} \gamma_{\alpha}^{P} D^{\alpha}$ and $Q(D) = \sum_{\alpha} \gamma_{\alpha}^{ Q} D^{\alpha}$ be linear differential operators, and let $P(\xi) = \sum_{\alpha} \gamma_{\alpha}^{P} \xi^{\alpha}$ and $Q(\xi) = \sum_{\alpha} \gamma_{\alpha}^{Q} \xi^{\alpha}$ be the corresponding symbols (characteristic polynomials). It is said that the operator $P(D)$ is more powerful than the operator $Q(D)$ (the polynomial $P(\xi)$ is more powerful than the polynomial $Q(\xi)$), and written $P>Q$ or $Q < P$, if there is a constant $c > 0$ such that $| Q(\xi)| \leq [| P(\xi) | + 1] \ \ \forall \xi \in \mathbb {R}^{n}$. In the talk, necessary and (in most cases coinciding) sufficient conditions are given under which $P > Q$, where $P$ is a degenerate (non-elliptic) operator. Using the obtained results and applying the theory of Fourier multipliers, we describe the set of multi-indices $\{\nu\}$ for which the estimate
$$\| D^{\nu} u \|_{L_{p}} \leq \| P(D)u \|_{L_{q}} + \| u \|_{L_{q}} \,\,\,\forall u \in C_{0}^{\infty}, 1 < p \leq q,$$
holds.