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.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.