(This question was previously posted on MSE and I decided to post it here too.)

I am studying the proof of the Malgrange preparation theorem given in the book "Stable mappings and their singularities" written by Golubitsky and Guillemin (see Chapter IV). The statement is the following.

Let $F\in C^\infty(\mathbb{R}\times\mathbb{R}^n; \mathbb{R})$ be defined on a neighborhood of $0\in\mathbb{R}\times\mathbb{R}^n$ and such that $F(t, 0) = g(t)t^k $ where $g(0)\neq0$ and $g$ is $C^\infty$ and defined on a neighborhood of $0\in\mathbb{R}$. Then there exist $C^\infty$ functions $q$, $\lambda_0$, ..., $\lambda_{k-1}$ such that $(qF)(t, x) = t^k + \sum_{i=0}^{k-1}\lambda_i(x)t^i$ on a neighborhood of $0\in\mathbb{R}\times\mathbb{R}^n$ and $q(0)\neq0$.

After a careful reading of the proof, it seemed to me that the result is preserved even if the assumption "$F\in C^\infty(\mathbb{R}\times\mathbb{R}^n; \mathbb{R})$" is replaced by "$F(t, \cdot)$ is $C^1$ for all $t$ and $F(\cdot, x)$ is $C^\infty$ for all $x$". (Obviously, the regularity of $q$, $\lambda_0$, ..., $\lambda_{k-1}$ must also be weakened in the same manner.) Indeed, all complex analysis arguments (inspired by Weierstrass preparation theorem) are used on functions depending on $t$, while only the implicit function theorem is used on a function depending on $x$.

My question is: Does anyone know a reference where this less regular version of the Malgrange preparation theorem is stated/proved/discussed?

Thank you.