Is there a real/functional analytic proof of Cramér–Lévy theorem?

Question

In the book Gaussian Measures in Finite and Infinite Dimensions by Stroock, there is a theorem with a comment

The following remarkable theorem was discovered by Cramér and Lévy. So far as I know, there is no truly probabilistic or real analytic proof of it.

Theorem 2.2.1 If $X$ and $Y$ are independent random variables whose sum is Gaussian, then each of them is Gaussian.

Let $\mathcal D$ be the set of all probability density functions (p.d.f.) on $\mathbb R$. Let $*$ denote convolution operation. We can re-phrase the theorem in the following functional analytic context, i.e.,

If $f, g \in \mathcal D$ such that $f*g$ is a Gaussian p.d.f., then $f, g$ are also Gaussian p.d.f.

Is there a real/functional analytic proof of my rephrased version?

Answer 1

I will just turn my comment into answer since this seems to still be an open problem as mentioned in "Regularized distributions and entropic stability of Cramer’s characterization of the normal law."

There they try to approach this question using entropy and prove Cramers for an example with noise but also give a counterexample.

The main purpose of this note is to give an affirmative solution to the problem in the (rather typical) situation, when independent Gaussian noise is added to the given random variables. That is, for a small parameter $\sigma > 0$, we consider the regularized random variables $X_{\sigma} = X + \sigma Z_1, Y_{\sigma} = Y + \sigma Z_2$, where $Z_1$ and $Z_2$ denote independent standard normal random variables, which are independent of $X, Y$.