Gagliardo-Nirenberg inequality with Hölder norms

Abstract

The classical Gagliardo-Nirenberg inequality, known as an interpolation inequality, involves Lebesgue norms of functions and their derivatives. We established an interpolation lemma to connect Lebesgue and Hölder spaces, thus extending the Gagliardo-Nirenberg inequality. This extension involved substituting arbitrary Sobolev norms with appropriate Hölder norms, allowing for a wider range of applicable parameters in the inequality.