A Sobolev inequality
It turns out that if 
is in 
, 
, where 
depend on the dimension. I have no idea how to prove this in general, and don’t really care.
BUT, it makes for a nice problem in the one-dimensional case. Show that if 
(i.e. its second derivative exists and both it and its first and second derivatives are square integrable), then 
, and find .
I haven’t a clue how to proceed, but it looks like mighty fun.
Update: the proof is, in retrospect (always in retrospect, damn it), obvious.
Possibly relevant posts:
- Distribution function (1/14/2006)
- Boundedness of products of certain matrices (10/29/2007)
- Product sigma algebras (9/26/2008)
For the 1d case, I used integration by parts on
, then cauchy-schwarz. Maybe you can do the same in higher dimensions.
Comment by Alex — 7/12/2008 @ 5:01 pm