Does hypoellipticity imply the existence of a parametrix?

Question

Let $M$ be a smooth manifold, like $\mathbb{R}^n$ for instance. The existence of a parametrix for an operator $P$ on $C^\infty(M)$ in any reasonable pseudodifferential calculus implies that $P$ is hypoelliptic. Is there a converse to this statement?

The converse would have to take place in some very general setting that encompasses all possible pseudodifferential calculi. Here's how I'd like to phrase it...

An operator $P$ on $C^\infty(M)$ is "very regular" if its Schwartz kernel has the following two properties:

• (1) $p$ is properly supported and semiregular in both variables, ie, $p \in (C^\infty(M) \hat\otimes \mathcal{E}'(M)) \cap (\mathcal{E}'(M) \hat\otimes C^\infty(M))$,

• (2) $p$ is equal to a smooth function off the diagonal.

The first condition means $P$ maps each of $C_c^\infty(M)$, $C^\infty(M)$, $\mathcal{E}'(M)$ and $\mathcal{D}'(M)$ to itself. The second means $P$ is pseudo-local.

Suppose a very regular operator $P$ is hypoelliptic, in the sense that every preimage of a smooth function is smooth. Does this mean that there is a very regular $Q$ which is a parametrix in the sense that $PQ-I$ and $QP-I$ are smoothing operators?

Answer 1

In the case of hypoelliptic operators with constant coefficients, you have a characterization found by L.Hörmander, which implies that you do have a pseudo-differential parametrix with a symbol in a class $S^{-m}_{\rho,\delta}$.

For hypoelliptic operators with variable coefficients, this is a different story: although the hypoelliptic operator in two dimensions $$ D_x^2+x^2 D_y^2, $$ does have a pseudo-differential parametrix, it is not so easy to put it in a classical framework of pseudo-differential operators. Moreover, the operator in three dimensions $$ D_x^2+x^4(D_y+x D_z)^2, $$ is hypoelliptic, but is not likely to have a parametrix with a symbol in any $\rho, \delta$ class and one can even doubt that a parametrix could live in a general class of pseudo-differential operators such as the ones created by R.Beals & C.Fefferman or L.Hörmander.