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.