Generalized Gagliardo–Nirenberg inequalities using Lorentz spaces, BMO, Hölder spaces and fractional Sobolev spaces

Abstract

The main purpose of this paper is to prove some generalized Gagliardo–Nirenberg interpolation inequalities involving the Lorentz spaces Lp,α, BMO and the fractional Sobolev spaces Ws,p, including also Ċη Hölder spaces. Although some of the results can be alternatively obtained by using interpolation spaces (specifically, the reiteration theorem), the precise form of the inequalities stated here appears to be novel and, moreover, the proofs given in the present paper are self-contained (save for the use of the John–Nirenberg inequality for the BMO result) in contrast to the other mentioned approach. The use of Ċη Hölder spaces in such Gagliardo–Nirenberg inequalities seems to be new in the literature.