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.