Interpolation between H\" older and Lebesgue spaces with applications

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.