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Let $D_1,D_2$ be two linear differential operators with matrix valued constant (i.e. translation invariant) coefficients on $\mathbb{R}^n$. Assume $D_2\circ D_1=0$, in other words $$Im(D_1)\subset Ker(D_2).$$

Let us assume that $Im(D_1)=Ker(D_2)$ for the class of polynomial (matrix valued) functions.

Under what conditions one may conclude that the same equality holds for $C^\infty$-smooth functions in a ball?

Basic examples of such situations (when the answer is known however) include:

  1. $D_1$ and $D_2$ are the Rham differentials $d$ on differential forms of appropriate degree.

  2. $D_1$ and $D_2$ are the Dolbeault differentials $\bar \partial$.

  3. $D_1=dd^c$ on real valued functions on $\mathbb{C}^n$, $D_2=d$ on real $(1,1)$-forms.

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    $\begingroup$ The exactness of your composition $D_2 \circ D_1=0$ in polynomials is essentially "formal exactness". Exactness in smooth functions is called "strict exactness" (in reference to this function class). For constant coefficient operators formal exactness implies strict exactness. Is your question essentially a duplicate of this earlier question? See my answer there for a reference to a detailed discussion. $\endgroup$
    – Igor Khavkine
    Commented Aug 3, 2022 at 19:34
  • $\begingroup$ @IgorKhavkine : Apparently the references there are what I need. Thanks a lot. $\endgroup$
    – asv
    Commented Aug 4, 2022 at 6:22
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    $\begingroup$ Does this answer your question? Reference request: Systems of linear PDES with constant coefficients $\endgroup$
    – Igor Khavkine
    Commented Aug 4, 2022 at 12:09
  • $\begingroup$ @IgorKhavkine: Yes. $\endgroup$
    – asv
    Commented Aug 4, 2022 at 12:12
  • $\begingroup$ Hmm, seems like a funny side effect of my vote to "close as a duplicate". $\endgroup$
    – Igor Khavkine
    Commented Aug 4, 2022 at 13:58

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