If the product of two functions is smooth, then how quickly must one function decay when the other is non-smooth? Suppose we have two functions $f,h$ on $\mathbb{R}$ such that:

• $h$ is Lipschitz continuous
• $f$ is smooth (i.e. $C^\infty$), and
• $fh$ is a smooth function.

What can we conclude about $f$ from this requirement? Presumably $f$ must go to zero 'sufficiently fast' at any point where $h$ is non-smooth.

For example, we can compute that a.e. $(fh)' = f'h + fh'$ and since $f'h$ is continuous, $h' = ((fh)' - f'h)/f$ is continuous in any neighborhood where $f$ is non-zero. Therefore $f$ is zero anywhere $h$ is not $C^1$. Must $f^{(k)}$ be zero anywhere $h$ is not locally smooth? Must $f^{(k)}h$ be smooth? Must $f^{(k)}h^{(\ell)}$ be well-defined (i.e. extend to be 0 at the set where $h$ is not $C^\ell$) and smooth?

In my application, $h$ is also smooth in an open, dense set, which may give some context though I doubt it helps with these.