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I must apologize for answering my own question since I should have checked the classical results in the literature before asking these two questions. However, let me summarize the situation: we always …

The hypoellipticity result is more precise: you have $$ Lu \in H^s_{loc}\Longrightarrow u\in H^{s+2-\delta}_{loc}\quad\text{ for some $\delta\in [0,2)$,} $$ and that $\delta$ is linked to the number o …

(1) Let me answer first to the last question: $\Delta \vert x\vert$ is homogeneous of degree $-1$ and radial. On $\mathbb R^d$ ($d\ge 2$) it is $$ (\partial_r^2+\frac{d-1}{r}\partial_r)(r)=\frac{d-1}{ …

Hormander's operator $L=X_0+\sum_{1\le j\le k} X_j^2$, where the $X_j$ are real smooth vector fields with the Lie algebra of $\{(X_j)\}_{0\le j\le k}$ generating the tangent space is hypoelliptic as …

Certainly, one should pay attention to the domain of the operator. However, the following argument should survive a reasonable assumption. We have $$ \frac{d}{dt}(E^\ast(t)E(t))=-2 E^\ast(t) A(t) E(t …