Verification of the Hormander condition I hope somebody can help me to understand how to verify the sufficient condition for hypoellipticity for the equation with $m$ variables? 
Hormander says: span of operators $X_k,[X_k,X_j],[X_k[X_j,X_l]]...$ spans the space, i.e. the rank of Lie algebra generated by the operators ${X_0,...,X_r}$ is equal to $m$. In his paper(1967) he provides an example of $$u_{xx}+xu_y-u_t=0$$ which satisfies that. I can't understand how to apply his sufficient condition in this case.  
 A: Following Hörmander's notation [see (1.6) below], let us write $u_{xx}+xu_y-u_t=(X_1^2+X_0)u$, where $X_1=\frac{\partial}{\partial x}$ and $X_0=x\frac{\partial}{\partial y}-\frac{\partial}{\partial t}$. We must check that at every point of $\mathbb R^3$ the vector fields $X_0,X_1$, and their commutators, have 3-dimensional span. This is immediate from $$[X_1,X_0]=\frac{\partial}{\partial x}\left(x\frac{\partial}{\partial y}-\frac{\partial}{\partial t}\right)-\left(x\frac{\partial}{\partial y}-\frac{\partial}{\partial t}\right)\frac{\partial}{\partial x}=\frac{\partial}{\partial y}$$
(Added) Let's check the spanning property. A set of vectors spans the ambient space (in this case 3 dimensional) iff the matrix of their components has maximal rank (ie, 3). Here we have $$\begin{pmatrix} 1&0&0\\0&x&-1\\0&1&0 \end{pmatrix} $$
which has rank 3 no matter what x is. Note that (a) both the original fields and their commutators are included; (B) in general the number of fields may be greater than the dimension, so the matrix is not necessarily square. 
(Remark) Hörmander does not distinguish the time derivative in his Theorem 1.1, and I followed his approach by putting it along with a space derivative into $X_0$. Noticing that the time derivative does not contribute to the commutators anyway, one may want to reformulate the Hörmander condition for parabolic equation so that the time derivative is left out of it completely. That is, we could write the equation $u_t=(X_1^2+\widetilde X_0)u$ where $X_1=\frac{\partial}{\partial x}$ and $\widetilde X_0=x\frac{\partial}{\partial y}$. Then we want $X_1$, together with all commutators of $\widetilde X_0$ and $X_1$, to span $\mathbb R^2$ at every point. I believe this is the correct parabolic form of the Hörmander condition, and the one given on Wikipedia is stated incorrectly (it appears to exclude $\widetilde X_0$ from participating in commutators, not just from the spanning set).  
For ease of reference, here is the main result of Hörmander's 1967 paper Hypoelliptic second order differential equations. 

