Interpolation between H\" older and Lebesgue spaces with applications

Abstract

Classical interpolation inequality of the type $\Vert u{\Vert }_X\le C\Vert u{\Vert }_Y^{\theta }\Vert u{\Vert }_Z^{1-\theta }$ 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 $\Vert \cdot {\Vert }_Y$ or $\Vert \cdot {\Vert }_X$ 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.

8 pages