Interpolation between H\" older and Lebesgue spaces with applications

TL;DR

This paper extends classical interpolation inequalities by incorporating Hölder semi-norms and weak Lorentz norms, leading to sharper results and broader applicability, including an extended Gagliardo--Nirenberg inequality.

Contribution

It introduces a generalized interpolation inequality replacing certain norms with Hölder semi-norms and weak Lorentz norms, enhancing the classical framework.

Findings

Extended interpolation inequalities with Hölder semi-norms.

Sharper inequalities involving weak Lorentz norms.

Broader Gagliardo--Nirenberg inequality applicability.

Abstract

Classical interpolation inequality of the type $\|u\|_{X}\leq C\|u\|_{Y}^{\theta}\|u\|_{Z}^{1-\theta}$ is well known in the case when $X$, $Y$, $Z$ are Lebesgue spaces. In this paper we show that this result may be extended by replacing norms $\|\cdot\|_{Y}$ or $\|\cdot\|_{X}$ by suitable H\" older semi-norm. We shall even prove sharper version involving weak Lorentz norm. We apply this result to prove the Gagliardo--Nirenberg inequality for a wider scale of parameters.