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.



Example of partial differential Operator that is analytic-hypoelliptic but not elliptic
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



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> *Bonfiglioli, Andrea; Lanconelli, Ermanno; Uguzzoni, Francesco*, [**Stratified Lie groups and potential theory for their sub-Laplacians**](http://dx.doi.org/10.1007/978-3-540-71897-0), Springer Monographs in Mathematics. New York, NY: Springer (ISBN 978-3-540-71896-3/hbk). xxvi, 800 p. (2007). [ZBL1128.43001](https://zbmath.org/?q=an:1128.43001).
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