Gagliardo-Nirenberg inequality with H\"older norms

TL;DR

This paper extends the classical Gagliardo-Nirenberg inequality by establishing a new interpolation lemma that connects Lebesgue and H"older norms, broadening its applicability.

Contribution

The authors introduce an interpolation lemma linking Lebesgue and H"older spaces, enabling the extension of the Gagliardo-Nirenberg inequality with H"older norms.

Findings

Extended Gagliardo-Nirenberg inequality with H"older norms

Broader parameter range for the inequality

New interpolation lemma connecting Lebesgue and H"older spaces

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 H\"older spaces, thus extending the Gagliardo-Nirenberg inequality. This extension involved substituting arbitrary Sobolev norms with appropriate H\"older norms, allowing for a wider range of applicable parameters in the inequality.