TITLE: Smallness of cut-off functions at critical Sobolev regularity
QUESTION [7 upvotes]: Consider the class of functions
$$X:=\{f\in \mathcal{C}_0^{\infty}(\mathbb{R})\;s.t.\;f\equiv 1  \mbox{ in a neighbourhood of}\;\;x=0\}$$
Is it true that, for every $\varepsilon > 0$, I can find $f\in X$ such that $\|f\|_{H^{1/2}(\mathbb{R})}<\varepsilon$?
For $s\in[0,1/2)$, it is easy to show that the analogous question has affirmative answer. Indeed, given $f\in X$ and $n>0$, define $f_n(x):=f(nx)$. Then $f_n\in X$, and $\|f_n\|_{H^{s}(\mathbb{R})}\to 0$ as $n\to +\infty$.
Instead, when $s>1/2$, the analogous question has negative answer. Indeed, by Sobolev embedding one has
$$\|f\|_{H^s}\geqslant C\|f\|_{L^{\infty}}\geqslant C|f(0)|= C$$
for any $f\in X$.
In the critical case $s=1/2$ I think the answer is affirmative, but I'm not able to prove it.
Thank you for any suggestion.

REPLY [4 votes]: It is well known and easy to verify that (Exercise 14 p. 309 in [1])
$$
\log\Big|\log\sqrt{x^2+y^2}\Big|\in H^1(B^2(0,e^{-1}))
$$
so the trace of this function on the $x$-axis belongs to the trace space
$$
f(x)=\log\Big|\log|x|\Big|\in H^{1/2}((-e^{-1},e^{-1})).
$$
Let
$$
f_t(x)=\begin{cases} 0 & \text{if } f(x)\leq t\\
f(x)-t & \text{if } t\leq f(x)\leq 2t\\
t & \text{if } f(x)\geq 2t
\end{cases}
$$
be a truncation of the function $f$ between the levels $t$ and $2t$, $t>0$. Then
$f_t\in H^{1,2}$ and $\Vert f_t\Vert_{1/2}\leq\Vert f\Vert_{1/2}$. 
Indeed, the space $H^{1/2}(\mathbb{R})$ is equipped with the norm
$$
\Vert u\Vert_{1/2}=
\Vert u\Vert_2+
\left(\int_{\mathbb{R}}\int_{\mathbb{R}}\frac{|u(x)-u(y)|^2}{|x-y|^2}\, dx\, dy\right)^{1/2}.
$$
Since $|f_t|\leq|f|$ and $|f_t(x)-f_t(y)|\leq |f(x)-f(y)|$, it immediately follows that $\Vert f_t\Vert_{1/2}\leq\Vert f\Vert_{1/2}$. Therefore
$$
\left\Vert\frac{1}{t}f_t\right\Vert_{1/2}\leq \frac{1}{t}\Vert f\Vert_{1/2}\to 0
\quad
\text{as $t\to\infty$.}
$$
The function $t^{-1}f_t$ equals $1$ near $0$ and it has compact support so approximating this function by convolution we can obtain a function $g_t\in C_0^\infty(\mathbb{R})$ such that $g_t=1$ near $0$ and $\Vert g_t\Vert_{1/2}<\varepsilon$, provided $t$ is sufficiently large.
[1] L. C. Evans, Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010.