Is there any continuous (and possibly bounded) function in the Sobolev space $\mathcal{H}^k(\mathbb{R}^d)$ but not in $\mathcal{H}^{d/2}(\mathbb{R}^d)$?
Where $k<d/2$ and $\mathcal{H}^k(\mathbb{R}^d)=\mathcal{W}^{k,2}(\mathbb{R}^d)$.
Since it seems quite standard to assume the Sobolev space with the smoothness bigger than $d/2$ because of the embedding result: if $k>d/2$, then $\mathcal{H}^k(\mathbb{R}^d)$ is continuously embedded in the bounded continuous functions $C_b(\mathbb{R}^d)$.
However, when the dimension $d$ is large and one only needs continuity, it is not convenient to assume the smoothness bigger than $d/2$. But also I need the function to be in $\mathcal{H}^k(\mathbb{R}^d)$ with some $k$ out of other reasons ($k<d/2$).
In other words, is it possible to prove
$C_b(\mathbb{R}^d) \cap \mathcal{H}^k(\mathbb{R}^d) \not\subseteq \mathcal{H}^{d/2}(\mathbb{R}^d)$ where $k<d/2$?
Any comments/references appreciated! (In the best case, I would like to see such function.)
The simplest examples will have $d = 4$, $d/2 = 2$, and $k=1$, so the question boils down to showing $$C_b(\mathbb R^4) \cap H^1(\mathbb R^4) \not\subseteq H^2(\mathbb R^4).$$
What you need is a function whose first order partial derivatives belong to $L^2$ but whose second order partial derivatives do not. You could try something like $$ \phi(x_1,x_2,x_3,x_4) = |x_1|^{2/3}$$ which is locally integrable and deal with the behavior at $\infty$ using a cutoff function $\eta$ with the property that $0 \le \eta \le 1$, $\eta(x) = 1$ for all $|x| \le 1$, and $\eta(x) = 0$ for all $|x| > 2$.
The function $f(x) = \phi(x) \eta(x)$ should do the trick.