Brezis-Gallouet-Wainger type inequality with critical fractional Sobolev space and BMO

In this paper, we prove the Brezis-Gallouet-Wainger type inequality involving the BMO norm, the fractional Sobolev norm, and the logarithmic norm of ${\dot{\mathcal{C}}}^{\eta }$, for $\eta \in (0,1)$.