Let $P(D)$ be hypoelliptic operator with constant coefficients in $\mathbb{R^n}$.Let $\Omega$ be an open subset of $\mathbb{R^n}$ and $\mathscr{N_\Omega}$ denote space of distribution solutions of homogeneous equation $P(D)h=0 $. I need to prove that following topologies on $\mathscr{N_\Omega}$ are identical:

(i) the $C^{\infty}$ topology (the uniform convergence of the functions and all their derivatives on every compact subset of $\Omega$),

(ii) the $C^0$ topology(the uniform convergence of the functions on every compact subset of $\Omega$),

(iii) the topology induced by $\mathscr{D'}(\Omega)$(the functions $f_{\alpha}\in \mathscr{N_\Omega}$ converge if the integrals $\int f_{\alpha} \phi dx$ converge for every test function $\phi \in \mathscr{D}(\Omega)$, uniformly on bounded subsets of $\mathscr{D}(\Omega)$ ).

We know that $\mathscr{N_\Omega} \in C^{\infty}(\Omega)$. Clearly $(i)\implies (ii) \implies (iii)$.

How do I see other way implications perticularly $(ii)\implies (i).$


1 Answer 1


There are at least three published proofs: The first (for the result in this generality -- for particular operators it is of course older) I know is of Malgrange [Existence et approximation des solutions des equations aux derivees partielles et des equations de convolution, Ann. Inst. Fourier, Grenoble, 6 (1955)–(1956), 271–355] using abstract results about strong duals of Frechet-Schwartz spaces, the second in Hörmander's book Analyiys of partial dillerential operators I (theorem 4.4.2) uses explicitely a fundamental solution with singular support equal to the origin, and a third one is in my article Topological properties of kernels of partial differential operators in Rocky Mountain J. Math. 44 (2014), 1037-1052.


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