Sharp Gagliardo-Nirenberg inequalities and mass transport theory

It is known that best constants and extremals of many geometric inequalities can be obtained via the Monge-Kantorovich theory of mass transport. But so far this approach has been successful for a special subclass of the Gagliardo-Nirenberg inequalities, namely, those for which the optimal functions involve only power laws. In this paper, we explore the link between Mass transport theory and all classes of the Gagliardo-Nirenberg inequalities. Sharp constants and optimal functions of all the Gagliardo Nirenberg inequalities are obtained explicitly in dimension n = 1, and the link between these inequalities and Mass transport theory is discussed.