# Equivalence of different topologies of Kernel of constant coefficient hypoelliptic operator

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

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 H?rmander'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.