Timeline for Sobolev embedding for fractional Sobolev spaces

7 events

when toggle format	what		by	license	comment
Sep 27, 2021 at 19:35	comment	added	Sahiba Arora		Sorry, I missed the part that it is sharp. Thank you!
Sep 27, 2021 at 19:28	comment	added	Mateusz Kwaśnicki		@SahibaArora: Sure, but what would you like to know about $u$? There's no hope for continuity (as indicated in my previous comment), so what could possibly serve as "something similar"?
Sep 27, 2021 at 19:22	comment	added	Sahiba Arora		Your answer considers $W^{\theta,2}(\Omega)$ for $\theta\in (1,2)$, I am interested in the case $\theta\in (0,1)$ with $n\geq 2$.
Sep 27, 2021 at 13:32	comment	added	Mateusz Kwaśnicki		@SahibaArora: I am afraid I do not know what do you mean "something similar". Hardy–Littlewood–Sobolev only gives $u \in L^{p^\star}$ with $p^\star=np/(n-\theta p)$, and this is sharp, I suppose. (Here $n \geqslant 2 > \theta p$ when $p = 2$. If, however, we allow $p$ to be greater than $n$, then, indeed, we get continuity of $u$ for $\theta$ large enough, namely, $\theta > n/p$.)
Sep 27, 2021 at 11:49	comment	added	Sahiba Arora		Can when say something similar for when $u\in W^{\theta,2}(\Omega)$ with $\theta\in (0,1)$ and $n\geq 2$?
Jan 13, 2021 at 14:07	vote	accept	Nirav
Jan 13, 2021 at 12:40	history	answered	Mateusz Kwaśnicki	CC BY-SA 4.0