Problem statement
Let $f,g \in C^\omega(X,\mathbb{R})$ be two real analytic functions over a real Banach space $X$.
Assume that, for every $n \in \mathbb{N}$, there exists $C_n>0$ and $h_n \in C^\omega(X,\mathbb{R})$ such that, for every $x \in X$, $$|f(x)-h_n(x) g(x)| \leq C_n \|x\|^{n+1}.$$
Does there exist a continuous function $h \in C^0(X,\mathbb{R})$ such that $f = hg$ in a neighborhood of $0$?
Some remarks
- When $g(0) \neq 0$, the result is straight-forward since it is classical that $f/g$ is locally real-analytic near $0$.
- When $g(0) = 0$, there is however no immediate obstruction to the result since the assumption with $n=0$ implies that $f(0) = 0$, and with $n=1$ implies that $\ker Dg_{|x=0} \subset \ker Df_{|x=0}$.
- One cannot hope for $f/g$ to be well-defined on the whole space $X$ since, for example, the assumption does not prevent the existence of some $x^* \neq 0$ such that $g(x^*) = 0$ but $f(x^*) \neq 0$.
- When $X = \mathbb{R}$, the result is straight-forward. Indeed, if $g$ is not identically $0$, then $g(x) = x^r \bar{g}(x)$ with $r \in \mathbb{N}^*$, $\bar{g} \in C^\omega(\mathbb{R},\mathbb{R})$ and $\bar{g}(0) \neq 0$. From the assumption with $n = r-1$, one checks that a similar decomposition holds for $f$. Hence $f/g$ is locally real-analytic as in 1.
- In my setting, $X$ is of infinite dimension. But I already do not know if the result holds or not when $X = \mathbb{R}^2$.
- In 1 and 4, one actually has $h \in C^\omega(X,\mathbb{R})$. In the general case, I would already be very happy with $h \in C^\infty(X,\mathbb{R})$ or even $h \in C^0(X,\mathbb{R})$.
The following references might be relevant, but I have been unable to settle my question using them. It seems that one knows characterizations of the polynomials $g$ for which the division by $g$ is well-defined within the ideal generated by $g$. But, in my setting, I can accept that the quotient has lower regularity than $f$ and $g$ (see 6).
- Vogt - A division theorem for real analytic functions, Bulletin of the London Mathematical Society, 2007
- Treves - Analytic PDES [Chapter 15, Division of Distributions], Grundlehren der mathematischen Wissenschaften, 2022
- Bierstone, Schwarz - Continuous linear division and extension of $C^\infty$ functions, Duke Mathematical Journal, 1983
I would very much appreciate any intuition, pointers or references on this problem!
