  We establish fine bounds for best constants of the fractional subcritical
Sobolev embeddings \begin{align*} W_{0}^{s,p}\left(\Omega\right)\hookrightarrow
L^{q}\left(\Omega\right), \end{align*} where $N\geq1$, $0<s<1$, $p=1,2$, $1\leq
q<p_{s}^{\ast}=\frac{Np}{N-sp}$ and $\Omega\subset\mathbb{R}^{N}$ is a bounded
smooth domain or the whole space $\mathbb{R}^{N}$. Our results cover the
borderline case $p=1$, the Hilbert case $p=2$, $N>2s$ and the so-called Sobolev
limiting case $N=1$, $s=\frac{1}{2}$ and $p=2$, where a sharp asymptotic
estimate is given by means of a limiting procedure. We apply the obtained
results to prove existence and non-existence of solutions for a wide class of
nonlocal partial differential equations.
