My question is about the Theorem 6.5 in the Hitchhiker's guide to the fractional Sobolev spaces.
Consider the case $n=1$ and $p=2$. Theorem 6.5 states that if $s<\frac12$, then $H^s(\mathbb R)$ is continuously embedded in $L^q(\mathbb R)$ for any $q\in [2, 2^*_s]$, where $2^*_s$ is the fractional Sobolev exponent.
What happens when $s=\frac12$? Does the continuous embedding hold anyway?
Possibly this result can not be deduced by Theorem 6.5, but maybe there is another way to include the case $s=\frac12$ in the embedding result (at least in my case of interest in which $n=1$ and $p=2$.
If it is the case, how one can prove it?
When $s=1/2$, we have that $H^s(\mathbb R) \hookrightarrow L^q(\mathbb R)$ for all $2\leq q<+\infty$ and the embedding fails for $q=+\infty$. All the proofs and detailing discussion can be found in Chapter 2 of “A first course in fractional Sobolev spaces” by Giovanni Leoni.