When the adjoint of a hypoelliptic operator hypoelliptic - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T07:36:35Z http://mathoverflow.net/feeds/question/85578 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/85578/when-the-adjoint-of-a-hypoelliptic-operator-hypoelliptic When the adjoint of a hypoelliptic operator hypoelliptic vkrouglov 2012-01-13T14:00:39Z 2012-03-05T14:49:19Z <p>Assume, $M$ is a smooth manifold with a measure $\mu$ and let $L^2(M, \mu)$ be a space of all square-integrable functions on $M$. </p> <p>Recall that $L$ is a <em>hypoelliptic differential operator</em>, if for every $f \in \mathcal{D}(L)$, if $Lf$ is in $C^\infty(M)$ then $f$ is also in $C^\infty(M)$.</p> <p>Could anyone give a reference to the example when $L$ is hypoelliptic but its adjoint w.r.t to $\mu$ is not hypoelliptic? Could this happen to the Hormander operators, when $L$ is defined as $$L = \sum_i X_i^2 + X_0$$ and ${X_i}$'s are bracket generating? </p> <p>I am mostly interested in the case when $\mu$ is induced by the Riemannian metric and $M$ is complete.</p> <p>Thanks,</p> http://mathoverflow.net/questions/85578/when-the-adjoint-of-a-hypoelliptic-operator-hypoelliptic/90286#90286 Answer by Bazin for When the adjoint of a hypoelliptic operator hypoelliptic Bazin 2012-03-05T14:49:19Z 2012-03-05T14:49:19Z <p>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 well as its adjoint since the Lie algebra condition does not change by taking adjoints.</p> <p>On the other hand, $\frac{\partial}{\partial t}+t\Delta_x$ is hypoelliptic whereas its adjoint $-\frac{\partial}{\partial t}+t\Delta_x$ is not hypoelliptic,</p> <p>Bazin.</p>