  The purpose of this article is twofold. The first is to strengthen fractional
Sobolev type inequalities in Besov spaces via the classical Lorentz space. In
doing so, we show that the Sobolev inequality in Besov spaces is equivalent to
the fractional Hardy inequality and the iso-capacitary type inequality.
Secondly, we will strengthen fractional Sobolev type inequalities in Besov
spaces via capacitary Lorentz spaces associated with Besov capacities. For this
purpose, we first study the embedding of the associated capacitary Lorentz
space to the classical Lorentz space. Then, the embedding of the Besov space to
the capacitary Lorentz space is established. Meanwhile, we show that these
embeddings are closely related to the iso-capacitary type inequalities in terms
of a new-introduced fractional $(\beta, p, q)$-perimeter. Moreover,
characterizations of more general Sobolev type inequalities in Besov spaces
have also been established.
