Upper bound of the best constant of the Trudinger-Moser inequality and its application to the Gagliardo-Nirenberg inequality Hideo KozonoTokushi SatoHidemitsu Wadade 46E3526D10Trudinger-Moser inequalityGagliadro-Nirenberg inequalitySobolev inequalityrearrangementaverage functionRiesz potentialfractional integral We will consider a Trudinger-Moser inequality for the critical Sobolev space $H^{n/p,p}(\mathbb{R}^n)$ with the fractional derivatives in $\mathbb{R}^n$ and obtain an upper bound of the best constant of such an inequality. Moreover, by changing normalization from the homogeneous norm to the inhomogeneous one, we will give the best constant in the Hilbert space $H^{n/2,2}(\mathbb{R}^n)$. As an application, we will obtain some lower bound of the best constant of a Gagliardo-Nirenberg inequality. Indiana University Mathematics Journal 2006 text pdf 10.1512/iumj.2006.55.2743 10.1512/iumj.2006.55.2743 en Indiana Univ. Math. J. 55 (2006) 1951 - 1974 state-of-the-art mathematics http://iumj.org/access/