Upper bound of the best constant of a trudinger-moser inequality and its application to A Gagliardo-Nirenberg inequality

By H. Kozono, Tokushi Sato, H. Wadade. 2004

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Fractional elliptic equations with critical exponential nonlinearity (2016, result): “The problems of type (P) with exponential growth nonlinearities are motivated from the fractional Trudinger–Moser inequality [18], which gives the optimal constant and improves the former results of Ozawa [26], and Kozono, Sato and Wadade [15].”
Remarks on Gagliardo–Nirenberg type inequality with critical Sobolev space and BMO (2008, unknown): “[10].”
On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form (2014, background): “Moreover, the inequality (1.9) in Corollary 1.2 was obtained as a particular case of the result in Nagayasu and Wadade [18, Corollary 1.3] which does not include the consideration for its sharp constant with respect to α.”
Sharp Adams type inequalities in Sobolev spaces Wm,nm(Rn) for arbitrary integer m (2012, background): “When β > β0, it is showed by Kozono, Sato and Wadade [19] and Proposition 6.”
Sharp Adams-type inequalities in ℝⁿ (2012, unknown): “([11], Corollary 1.”
Sharp singular Adams inequalities in high order Sobolev spaces (2011, background): “It was proved in [26] that the inequality (1.”
A new approach to sharp Moser–Trudinger and Adams type inequalities: A rearrangement-free argument (2013, methodology): “To show the sharpness of β0(n, γ ), we proceed as in [2,28,27,48].”