Timeline for Interior of positive cone in Sobolev space

5 events

when toggle format	what		by	license	comment
Dec 17, 2015 at 9:39	comment	added	Piero D'Ancona		Just by rescaling. The norm of $f_a$ is a power of $a$ times the norm of $f$. When $a\to0$ the behaviour is the one written above
Dec 16, 2015 at 22:03	comment	added	Dee		Sorry, now that I've written it down, I don't see how sobolev norm of $f_a$ is calculated?
Dec 16, 2015 at 15:14	comment	added	Piero D'Ancona		Take a test function $f(x)\ge0$, $0<s<-1+n/p$, and let $f_a(x)=a^{-s}f(x/a)$ for $a>0$. The Sobolev norm of $f_a$ is $\simeq a^{-s-1+n/p}\to0$ as $a\to0$, while the sup norm of $f_a$ tends to $\infty$. Thus we can modify locally the sign of a function by adding or subtracting a perturbation $f_a$ arbitrarily small in the Sobolev norm.
Dec 16, 2015 at 13:51	review	First posts
Dec 16, 2015 at 13:54
Dec 16, 2015 at 13:42	history	asked	Dee	CC BY-SA 3.0