Let $\Omega\subset \mathbb R^n$ be a convex subset. All the objects below will be defined on this set.

Let us assume $P(x,D)$ to be a differentiable operator order $m$ and of square size, that is having as many equations as unknowns, with variable smooth coefficients.

Let moreover $Q(D)$ another differentiable operator of order $m$ and of square size (the same of $P$), but this time having **constant** coefficients.

We also assume that

- $P$ is of constant strength in the sense of Hormander, that is for any $x,y\in \Omega$ there exist a constant $C_{xy}$ such that $$|P(x,\xi)|/|P(y,\xi)|\leq C_{xy},$$ for any $\xi\in\mathbb R^n$
- The operators $P$ and $Q$ are of equal strength, that is for any $x\in\Omega$ (it is not anymore important who is $x$ by the previous point), we have that there exist a constant $C$ such that $$\frac{1}{C}\leq\frac{|P(x,\xi)|}{|Q(\xi)|}\leq C,\quad\forall \xi\in\mathbb R^n$$

I want to solve now the equation $P(x,D)u=f$, for $f=(f_1,\dotso,f_k)\in (C^\infty_c(\Omega))^k$, and of course deduce something on the regularity of the solution $u$.

If I understand well, the heuristic should be as follows:

- The equation $Q(D)u=f$ admits a smooth solution $u$ (Ehrenpreis fundamental principle), this should be the hard part to prove, but I take it as a black box
- Since $P$ and $Q$ are of equal strength, by the previous hypotheses, then the solution $u$ to $P(x,D)u=f$ has to be "as regular as" the solution $v$ to $Q(D)v=f$. In particular I expect $u$ to exist and to be smooth, well, even compactly supported since so is $f$. This is in fact the guiding principle of Hormander when he introduce this concept of strength as a preorder relation between operators in
*Linear partial differential operators, chap. 3*.

Is my heuristic correct? How can one be rigorous in proving this? Thank you very much in advance