Interpolation between H\" older and Lebesgue spaces with applications

Classical interpolation inequality of the type $|u|{X}\leq C|u|{Y}^{\theta}|u|_{Z}^{1-\theta}$iswellknowninthecasewhen$X$,$Y$,$Z$ are Lebesgue spaces. In this paper we show that this result may be extended by replacing norms $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_Y$ or $\mathrm{\parallel }\cdot {\mathrm{\parallel }}_X$ 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.