TITLE: On Glaeser's Theorem for non-smooth functions
QUESTION [8 upvotes]: Glaeser's Theorem says that a $C^\infty$ function $F$ on $\mathbb R^n$ which is invariant under permutation of the variables is a smooth function of the symmetric polynomials of $(x_1, \dots, x_n)$. 
Question 1: What remains (if anything) of this statement if $F$ is $C^k$ ?
Question 2: In the statement above, is it clear that you can replace $\mathbb R^n$ by a symmetric open subset of $\mathbb R^n$?

REPLY [9 votes]: A more direct reference (quoted in Rumberger's paper) is: G. Barbançon, Le théorème de Newton pour les fonctions de classe $C^r$. Ann. Sci. École Norm. Sup. 5 (1972), 435–458. He proves that a symmetric function  of class $C^{nr}$ on $\Bbb{R}^n$ is a function  of class $C^{r}$ of the symmetric polynomials.