The aim of this paper is to give a mass transportation proof for a full family of sharp Gagliardo–Nirenberg inequalities in dimension one. In fact, we shall establish a duality principle which derives this family of inequalities as a consequence. We also characterize all optimizers for these inequalities via the mass transportation method.
"A mass transportation proof of the sharp one-dimensional Gagliardo–Nirenberg inequalities." J. Math. Soc. Japan 73 (2) 633 - 647, April, 2021. https://doi.org/10.2969/jmsj/82258225