Consider the differential operator $P$ on $\mathbb{R}^2$, given by $P = \frac{\partial^2}{\partial x^2} + x^2\frac{\partial^2}{\partial y^2}$. Clearly it is elliptic everywhere except on the $y$-axis. I am looking to make a statement like $P\varphi \in H^s_{\text{loc}} \Rightarrow \varphi \in H^{s + r}_{\text{loc}}$. Clearly, $r \leq 2$, but how big can $r$ be? I was told by someone that Hörmander's paper titled "A class of hypoelliptic pseudodifferential operators with double characteristics" (Math. Ann. 217 (1975), no. 2, 165–188) deals with problems of this nature, but the paper is a bit too technical for me. Anyone, any insights? Or any other references that are more easily readable? Thanks a lot in advance.

Edit: I mentioned this particular operator $P$ just as an example. Clearly, one could construct many others, for example, $y^2\frac{\partial^2}{\partial x^2} + x^2\frac{\partial^2}{\partial y^2}$ and ask this question on compact manifolds too. As pointed out below, these are definitely hypoelliptic. And, as Sönke Hansen points out, the loss should be at most one, but I cannot prove it.


You have $P=X_1^2+X_2^2$ with the real vector fields $X_1=\partial_x$ and $X_2=x\partial_y$. Since $X_1$ and the commutator $[X_1,X_2]$ generate all vector fields, Hörmander's theorem applies and gives hypoellipticity with loss of $2-\varepsilon$ derivatives. Chapter 9 of A. Unterberger, Pseudo-differential operators and application, Aarhus Universitet Lecture Notes Series No.46, 1976, and Hörmander III.22.2 contain proofs of the theorem.

More precise results about the loss of derivatives were shown by Folland, Rothschild, and Stein using approximation by nilpotent Lie groups, see MR0436223. For the $P$ above, hypoellipticity with loss of one derivative holds. This follows from Theorem 18 in the Rothschild-Stein paper (MR number given above) since the weight, defined in that paper, equals $2$ for $P$.

  • $\begingroup$ Can you please give me a reference which lists sufficient conditions for the loss to be at most one? $\endgroup$ – mathenthusiast May 27 '15 at 13:47
  • $\begingroup$ Thanks for your reply and I looked at the paper you mentioned. I am still having trouble though understanding exactly what manifolds $M$ the findings of Theorem 16 and 18 apply. Do they apply, for instance, to a sphere? I would be really grateful for some insight on this. $\endgroup$ – mathenthusiast Jun 17 '15 at 13:57

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