Second derivative seminorm of function composition - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-18T07:52:14Z http://mathoverflow.net/feeds/question/94493 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/94493/second-derivative-seminorm-of-function-composition Second derivative seminorm of function composition John Schulman 2012-04-19T04:47:30Z 2012-04-19T07:57:51Z <p>Consider the seminorm $\| f \|^2 = \int_{-\infty}^{\infty} dx f''(x)^2$ </p> <p>for $f:\mathbb{R}\rightarrow \mathbb{R}$ in the Sobalev space $W^{k,2}(\mathbb{R})$.</p> <p>Can we put some upper bound on the composition $\| f\circ g \|$ in terms of $\| f \|$ and $\| g \|$?</p> http://mathoverflow.net/questions/94493/second-derivative-seminorm-of-function-composition/94505#94505 Answer by Dirk for Second derivative seminorm of function composition Dirk 2012-04-19T07:57:51Z 2012-04-19T07:57:51Z <p>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 $\lambda^6$.</p>