Second derivative seminorm of function composition

Question

Consider the seminorm $\| f \|^2 = \int_{-\infty}^{\infty} dx f''(x)^2$

for $f:\mathbb{R}\rightarrow \mathbb{R}$ in the Sobalev space $W^{k,2}(\mathbb{R})$.

Can we put some upper bound on the composition $\| f\circ g \|$ in terms of $\| f \|$ and $\| g \|$?

Answer 1

I don't think so. Consider the rescaling $f_\lambda(x) = f(\lambda x)$ and $g_\lambda(x) = g(\lambda x)$. Then the term $\|f_\lambda\circ g_\lambda\|^2$ scales like $\lambda^7$ while $\|f_\lambda\|^2\|g_\lambda\|^2$ scales like $\lambda^6$.