Nonlinear Analysis 173 (2018) 146–153 Contents lists available at ScienceDirect Nonlinear Analysis www.elsevier.com/locate/na Generalized Gagliardo–Nirenberg inequalities using Lorentz spaces, BMO, Hölder spaces and fractional Sobolev spaces Nguyen Anh Dao a, * , Jesus Ildefonso Díaz b , Quoc-Hung Nguyen c a Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Viet Nam b Instituto de Matemática Interdisciplinar, Universidad Complutense de Madrid, 28040 Madrid, Spain c Scuola Normale Superiore, Centro Ennio de Giorgi, Piazza dei Cavalieri 3, I-56100 Pisa, Italy article info Article history: Received 26 September 2017 Accepted 1 April 2018 Communicated by Enzo Mitidieri Keywords: Gagliardo–Nirenberg’s inequality Lorentz spaces BMO Lipschitz space Fractional Sobolev space abstract The main purpose of this paper is to prove some generalized Gagliardo–Nirenberg interpolation inequalities involving the Lorentz spaces L p,α , BMO and the fractional Sobolev spaces W s,p , including also ˙ C η 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 ˙ C η Hölder spaces in such Gagliardo–Nirenberg inequalities seems to be new in the literature. © 2018 Elsevier Ltd. All rights reserved. 1. Introduction The main purpose of this paper is to prove some generalized Gagliardo–Nirenberg interpolation inequalities involving the Lorentz spaces L p,α , BMO, and the fractional Sobolev spaces W s,p , including also ˙ C η H¨ older spaces. It is well known that the Gagliardo–Nirenberg inequality plays an important role in the analysis of PDEs, see e.g. [10,8,9,1,11,7,3] and the references therein. Thus, any possible improvement of this one could be relevant for many purposes. First of all, let us recall some previous results involving the Gagliardo–Nirenberg inequalities that we shall improve later: For any 1 ≤ q<p< ∞, the following interpolation inequality holds (see Nirenberg [10]) ∥f ∥ L p (R n ) ≤ c∥f ∥ θ L q (R n ) ∥f ∥ 1−θ ˙ H s (R n ) , 1 p = θ q + (1 − θ)( 1 2 − s n ). (1.1) * Corresponding author. E-mail addresses: daonguyenanh@tdt.edu.vn (N.A. Dao), ildefonso.diaz@mat.ucm.es (J.I. D´ ıaz), quoc-hung.nguyen@sns.it (Q.-H. Nguyen). https://doi.org/10.1016/j.na.2018.04.001 0362-546X/© 2018 Elsevier Ltd. All rights reserved.