Remarks on the fractional Moser-Trudinger inequality

In this article, we study the connection between the fractional Moser-Trudinger inequality and the fractional $(\frac{kp}{p-1},p)$-Poincar'e type inequality for any Euclidean domain and discuss the sharpness of this inequality whose analogous results are well known in the local case. We further provide sufficient conditions on domains for fractional $(q,p)$-Poincar'e type inequalities to hold. We also derive Adachi-Tanaka type inequalities in the non-local setting.