On Glaeser's result for the square-root of a smooth non-negative function

One of the results due to Georges Glaeser is the following: there exists a non-negative $$C^\infty$$ function $$f$$ on the real line, flat at its zeroes, such that $$\sqrt{f}$$ is not $$C^2$$. On the other hand, $$\sqrt{f}$$ is $$C^1$$ for any such $$f$$.

Question 1: is it possible to find $$f$$ as above such that $$\sqrt{f}$$ is not twice differentiable at a point? In Glaeser's counterexample mentioned above, $$\sqrt{f}$$ is twice differentiable with an unbounded second derivative.

Question 2: is it possible to find $$f$$ as above such that there is no function $$g$$, $$C^2$$ on the real line such that $$f=g^2$$. Here $$g$$ is allowed to take negative values, which is not the case of $$\sqrt f$$.

Question 3: is it possible to find $$f$$ as above such that there is no $$C^\infty$$ function $$g:\mathbb R\longrightarrow \mathbb C$$ such that $$f=\vert g\vert^2$$.

Question 1: In

• Alekseevsky, Dmitri; Kriegl, Andreas; Michor, Peter W.; Losik, Mark Choosing roots of polynomials smoothly. (English summary) Israel J. Math. 105 (1998), 203–233.

it is shown that such an $$f$$ always has a twice differentiable square root. But this square root is not necessarily positive.

Question 2:

$$f(t) = \sin^2(1/t)e^{-1/t} + e^{-2/t} \text{ for }t>0,\quad f(t) = 0\text{ for }t\le0.$$ This is a sum of two non-negative $$C^\infty$$ functions each of which has a $$C^\infty$$ square root. But the second derivative of the square root of $$f$$ is not continuous at the origin. This is also a counter example to question 3.

• J.-M. Bony, F. Broglia, F. Colombini and L. Pernazza, Nonnegative. functions as squares or sums of squares, J. Funct. Anal. 232(2006), 137–147.

have shown that twice differentible is best possible; it cannot be improved to $$C^{1,\alpha}$$ for any continuity module $$\alpha$$.