Timeline for Understanding spaces of negative regularity

9 events

when toggle format	what		by	license	comment
Jul 17 at 15:23	comment	added	Pietro Majer		Yes, indeed I've never seen it before
Jul 17 at 5:43	comment	added	Martin Hairer		@PietroMajer But don't forget that as mentioned above there is no natural convention under which $C^{-k}$ is the dual of $C^k$.
Jul 17 at 4:42	comment	added	CBBAM		@PietroMajer Thank you.
Jul 17 at 4:15	comment	added	Pietro Majer		As $-D^:(C^{k})^\to (C^{k+1})^*$...
Jul 17 at 1:44	comment	added	CBBAM		@PietroMajer Sorry I did not fully understand the last part of your comment. Would you be able to explain how you get $C^{-k}\to C^{-k-1}$?
Jul 17 at 1:43	comment	added	CBBAM		@MartinHairer Thank you very much!
Jul 16 at 20:55	comment	added	Pietro Majer		I’d say the reason (say in one variable) relies in the fact that, first of all, the distributional derivative takes $C^0(I)$ to its dual, and for the rest, it is defined as $-D^*$, so that from the derivative $C^{k+1}\to C^k$ you still get a derivative $C^{-k}\to C^{-k-1}$.
Jul 16 at 20:42	comment	added	Martin Hairer		I would recommend you read through Section 14.3 of hairer.org/notes/RoughPaths.pdf which answers your questions (and more). It's five pages, it's self-contained (except maybe for a couple of definitions), and it uses only the definition of distributions and elementary calculus. Note that $C^{-k}$ is quite different from the dual of $C^k$ (think of $k=0$)...
Jul 16 at 19:58	history	asked	CBBAM	CC BY-SA 4.0