What are the subelliptic estimates for the Rockland operator?

Question

Let $X=(X_1, X_2, \dots, X_n)$ be a smooth vector field on $\mathbb{R}^n$. The operator $L=(\sum_{i=1}^{m}X_i^2)^p$, where $p$ is an integer, is a degenerated operator. If $X$ satisfies the Hörmander condition, for the case of $p=1$, we have the subelliptic estimates

$$|u|_{s+a}\leq |Lu|_{s} + |u|_{L^2}$$

for $u \in L^2(\Omega)\cap C^{\infty}_0$; here $|\cdot|_{s}$ denotes the Sobolev norm.

Now I want to know what if $p\geq 2$? Moreover, what about general case? i.e. Rockland operator (i.e. higher order operator). I want to get the estimate like the case $p=1$.

Answer 1

Let us set $L_1=\sum_{1\le j\le m}X_j^2$, where the $X_j$ are smooth real vector fields satisfying Hörmander's condition so that $L_1$ is subelliptic with an estimate $$ \Vert L_1 u\Vert_{H^s}\ge C\Vert u\Vert_{H^{s+\frac{2}{k+1}}}, $$ for any $u\in C^\infty_c$ with a diameter of the support small enough included in a fixed compact set. We define $L_p=(L_1)^p$; let us try our hand on $L_2$. We have $$\Vert L_2 u\Vert_{H^s}=\Vert L_1 L_1 u\Vert_{H^s}\ge C\Vert L_1 u\Vert_{H^{s+\frac{2}{k+1}}}\ge C^2 \Vert u\Vert_{H^{s+\frac{4}{k+1}}}. $$ Similarly, we have $$ \Vert L_p u\Vert_{H^s}\ge C^p\Vert u\Vert_{H^{s+\frac{p}{k+1}}}. $$