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.



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> **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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Sobolev embedding for fractional Sobolev spaces
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 $\theta > 0$.


