@TerryTao For the Tychonoff counterexample for the heat equation, the non-null solution increases as $e^{\vert x\vert^2}$ and is not a tempered distribution. To recover uniqueness, it is in fact enough to assume that the solutions live in the space of tempered distributions, a rather mild assumption. You are right the $L^r_{t,x}$ bound is a global one. If you ask for the big problem of uniqueness of Leray solutions (in $L^\infty_tL^2_x\cap L^2_t\dot H^1_x$) which are also smooth, it is not clear that you have ended-up with a much simpler problem although you have ruled out turbulent solutions.

@Terry Tao Assuming $r\ge 2$ or that $v,w$ are both Leray solutions, my point is that the great problem of uniqueness is not simplified when the solutions are smooth (a local property); in fact to apply the standard uniqueness result stated in my question, you need a global property $L^p_tL^q_x$.

Thank you very much. I have changed slightly the formulation.

Is there not a "Schwartz Kernel Theorem" in that context saying that all linear bounded operators must have a (distribution) kernel?

@Bob Yuncken If you take a look at the "lifting" procedure due to Rothschild \& Stein to prove the optimal hypoellipticity of the second operator, you will see that this is very far from the explicit construction of a parametrix. Also if you look at Chapter 27 on Subellipticity in L. Hörmander's ALPDO volume IV, you will check that the subellipticity of $D_x+i x^2(D_y+ c\vert D_z\vert)$ requires the full strength of the whole chapter which does not lead to a parametrix construction.

Very nice argument, thanks.

Thanks for your example: you are certainly right that oscillatory behaviour should be the key of this question, but I guess that for your function the series of $c_n$ is absolutely convergent (as $e^{-n}$).

@Christian Remling: I want to find a $C^\infty$ function $f$ such that, with $g=\vert f\vert$, the distribution second derivative of $g$ is not a Radon measure.