Example of partial differential Operator that is analytic-hypoelliptic but not elliptic We know that heat equation is hypoelliptic but not analytic-hypoelliptic; and also operators such as Cauchy-Reimann,laplacian etc are elliptic. I would like to know what operator is analytic-hypoelliptic but not elliptic.
 A: A canonical example is the sub-Laplacian $L$ on the real 3-dimensional Heisenberg group $\mathbb{H}^3$.  If we realize the Heisenberg group as $\mathbb{R}^3$, we can write $L = X^2 + Y^2$ where
$$X = \frac{\partial}{\partial x} - \frac{1}{2} y \frac{\partial}{\partial z}, \quad Y = \frac{\partial}{\partial y} + \frac{1}{2} x \frac{\partial}{\partial z}.$$
You can check that this fails to be elliptic (basically, because it is the sum of the squares of two linearly independent vector fields in a three-dimensional space).  But it is analytic-hypoelliptic; one can write down the fundamental solution (Green's function) explicitly and see that it is real analytic away from the diagonal.  A key fact is that $X,Y$ together with their Lie bracket $[X,Y] = \frac{\partial}{\partial z}$ span the tangent space at each point; this is Hörmander's bracket generating condition.
You can find details, references and other examples in Section 5.10 of

Bonfiglioli, Andrea; Lanconelli, Ermanno; Uguzzoni, Francesco, Stratified Lie groups and potential theory for their sub-Laplacians, Springer Monographs in Mathematics. New York, NY: Springer (ISBN 978-3-540-71896-3/hbk). xxvi, 800 p. (2007). ZBL1128.43001.

