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

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