__This is an exercise of Introduction to the Theory of Distributions by Friedlander and Joshi.
(I) Let $I \subset \mathbb{R}$ be an open interval. Show that if $u \in \mathcal{D}'(I)$ and $\partial u \in C^{\infty}(I)$, then $u \in C^{\infty}$.
(II) Further, let $P$ be a differential operator with smooth coefficients of order $m$, i.e. $$P(x,\partial)=\sum_{0\le k\le m} a_k(x)\partial^k,\quad a_k \in C^\infty(I)$$ whose leading coefficient $a_m(x)$ never vanishes on $I$. Show that if $u \in \mathcal{D}'(I)$ and $P u \in C^{\infty}(I)$, then $u \in C^{\infty}$.
__The authors gave a hint, stated as
Assume that, for each $x_0 \in I$, there is a $\phi \in C^\infty(I)$ such that $P\phi =0 $ and $\phi(x_0)=1$, and use induction. The existence of such $\phi$ follows from the theory of ODEs.
__I know that this result follows directly from the elliptic regularity of elliptic pseudodifferential operators. But I also want to solve this problem without the help of the theory of PsDOs.
__The first problem is easy. Since $v=\partial u$ is smooth, we may find a primitive $w \in C^\infty(I)$ for $v$, i.e. $\partial w=v$. Then we see that $w$ agrees $u$ on the subspace $$\partial C_c^\infty(I) :=\{ \psi \in C_c^\infty(I)|\exists \phi \in C_c^\infty(I),\psi=\partial \phi\},$$ which is also identified with the kernel of integration $$\partial C_c^\infty(I)=\{ \psi|\int_I \psi =0\}.$$
Fix a function $\rho \in C_c^\infty$ such that $\langle 1,\rho \rangle = \int_I \rho =1$, and for each $\phi \in C_c^\infty(I)$ we can write $$\phi = \phi - \langle 1,\phi \rangle\rho + \langle 1,\phi \rangle\rho.$$ Then we can see that $\phi-\langle 1,\phi \rangle \rho \in \partial C_c^\infty(I)$. Thus $$\langle u,\phi \rangle = \langle u,\phi-\langle 1,\phi \rangle \rho \rangle + \langle 1,\phi \rangle \langle u,\rho \rangle = \langle w,\phi-\langle 1,\phi \rangle\rho\rangle+\langle c,\phi \rangle=\langle w+C,\phi \rangle,$$ and $u=w+C \in C^\infty(I).$
__Now for the second one, W.L.O.G we may assume that the leading term is constant $1$ and start with the simplest case $$P=\partial + a_0 u.$$ But even in this case I cannot make any further progresses. It seems that the above argument can be applied similarly to the second problem, but unfortunately I cannot find such a simple characterization for $P^TC_c^\infty(T)$, which is important in the first solution. Moreover, I failed to understand what the hint trying to tell, and how should the induction be performed.
__Thanks for reading so far!