Reference Request for a theorem on translation invariant operators on $C_c^\infty(\mathbb{R}^n)$

Question

I think a result with possible minor modifications in the hypothesis should be true. I am looking for a reference for such a result. Any leads are greatly appreciated.

Let $\Lambda:C^\infty_c(\mathbb{R}^n)\longrightarrow C^\infty_c(\mathbb{R}^n)$ be a translation invariant continuous linear operator i.e. $T_z\Lambda T_z^{-1}=\Lambda$ for all $z\in \mathbb{R}^n$ where $T_z f(x)=f(x-z)$ for all $f\in C^\infty_c(\mathbb{R}^n)$. Then $\Lambda$ is given by convolution with a distribution i.e. there exists $\lambda\in (C^\infty_c(\mathbb{R}^n))'$ such that $\Lambda(f)=\lambda *f$ for all $f\in C^\infty_c(\mathbb{R}^n)$.

For context, this point came up in proving adjointability for a particular case of pseudodifferential operators.

Answer 1

Seems like Walter Rudin's Functional Analysis, Chapter $6$, Theorems $6.33$ and $6.35$ (pages $173$ to $175$) have got you covered. Alternatively, another reference is William Francis Donoghue Jr.'s Distributions and Fourier Transforms, pages $121$ and $122$ (no numerotation on the theorems).

In both books, $\mathscr{D}$ or $\mathscr{D}(\mathbb{R}^n)$ is the notation for $\mathcal{C}^\infty_c(\mathbb{R}^n)$. Those two books even first generalise your result to operators $L : \mathscr{D} \to \mathcal{C}^\infty$ (and even $L : \mathscr{D} \to \mathcal{C}(\mathbb{R}^n)$ in the case of Rudin, but he shows that if $L$ is invariant then the range actually lied in $\mathcal{C}^\infty$ from the get-go!) before taking your particular case, in which case the distribution that appears for the convolution is compactly supported.

I just found these references through an article titled Linear operators commuting with translations on $\mathscr{D}(\mathbb{R})$ are continuous by Gary Hosler Meisters which proves that, well, like the title suggests, linear operators which are translation invariant from $\mathscr{D}(\mathbb{R})$ to itself are continuous through an application of a generalisation of the Closed Graph Theorem, meaning that the continuous hypothesis is superfluous and the continuity of the operator is actually a given. Which is a pretty neat result I dare say.

Hopefully that helped.