Interpolation between Hölder and Lebesgue spaces with applications

Classical interpolation inequality of the type ‖u‖X≤C‖u‖Yθ‖u‖Z1−θ 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 ‖⋅‖Y or ‖⋅‖X by suitable Hölder 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.