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Let $\Omega\subset\mathbb{R}^2$ be open and of class $C^1$. The Sobolev embedding theorem implies that if $u\in W^{k,2}(\Omega)$ and if $k\in\mathbb{N}: k\geq 2$, then $u$ is continuous.

Question. Does there exist a similar result for fractional Sobolev Spaces? For example, if $u\in W^{1+\theta,2}(\Omega)$ for some $\theta\in (0,1)$, then can we say that $u$ is continuous?

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  • $\begingroup$ That looks like (a variant of) the Hardy–Littlewood–Sobolev inequality. Have you searched for this result in Stein's book? Another obvious place to look into is Hitchhiker's guide to the fractional Sobolev spaces. Or Samko's book on hypersingular integrals. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jan 13, 2021 at 12:00
  • $\begingroup$ @MateuszKwaśnicki Thank you for the references. I'll have a look. I found something close in Hitchhiker's guide to the fractional Sobolev Spaces (Thm. 8.2), but it's not exactly what I'm after. $\endgroup$
    – Nirav
    Commented Jan 13, 2021 at 12:18
  • $\begingroup$ @Francesco, 14 edits of (mostly) old questions in a few hours, pushing newer questions off the front page. Please don't do that. $\endgroup$
    – Gerry Myerson
    Commented Feb 19, 2021 at 11:59

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If $\Omega$ is a "nice" domain in $\mathbb R^n$ and $u \in W^{1+\theta,p}(\Omega)$ with $\theta \in (0, 1)$, then both $u$ and the weak gradient $\nabla u$ are in $W^{\theta,p}(\Omega)$, and hence, by the Hardy–Littlewood-Sobolev inequality (Theorem 6.7 in the Hitchhiker's guide to the fractional Sobolev spaces), $u$ and $\nabla u$ are in $L^{p^\star}(\Omega)$, with $p^\star = n p / (n - \theta p)$. It follows that $u$ is in $W^{1,p^\star}$, and consequently, by the usual Sobolev inequality, $u$ is continuous whenever $p^\star > n$. This boils down to $p > n - \theta p$, or $\theta > \tfrac np - 1$.

In your case, $n = p = 2$, so one only needs $\theta > 0$.

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  • $\begingroup$ Can when say something similar for when $u\in W^{\theta,2}(\Omega)$ with $\theta\in (0,1)$ and $n\geq 2$? $\endgroup$
    – Sahiba Arora
    Commented Sep 27, 2021 at 11:49
  • $\begingroup$ @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, $\theta > n/p$.) $\endgroup$
    – Mateusz Kwaśnicki
    Commented Sep 27, 2021 at 13:32
  • $\begingroup$ Your answer considers $W^{\theta,2}(\Omega)$ for $\theta\in (1,2)$, I am interested in the case $\theta\in (0,1)$ with $n\geq 2$. $\endgroup$
    – Sahiba Arora
    Commented Sep 27, 2021 at 19:22
  • $\begingroup$ @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"? $\endgroup$
    – Mateusz Kwaśnicki
    Commented Sep 27, 2021 at 19:28
  • $\begingroup$ Sorry, I missed the part that it is sharp. Thank you! $\endgroup$
    – Sahiba Arora
    Commented Sep 27, 2021 at 19:35

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