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.