Fractional Adams–Moser–Trudinger type inequalities. (November 2015)

• Record Type: Journal Article
• Authors: Martinazzi, Luca
• Is Part Of: Nonlinear analysis. Volume 127(2015)
• Page Start: 263
• Page End: 278
• Publication Date: 2015-11
• Subjects: Nonlocal equations -- Moser–Trudinger inequalities -- Fractional Sobolev spaces; Mathematical analysis -- Periodicals; Functional analysis -- Periodicals; Nonlinear theories -- Periodicals; Analyse mathématique -- Périodiques; Analyse fonctionnelle -- Périodiques; Théories non linéaires -- Périodiques; Functional analysis; Mathematical analysis; Nonlinear theories; Periodicals; Electronic journals; 515.7248
• DOI: 10.1016/j.na.2015.06.034
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: Extending several works, we prove a general Adams–Moser–Trudinger type inequality for the embedding of Bessel-potential spaces H ̃ n p, p ( Ω ) into Orlicz spaces for an arbitrary domain Ω with finite measure. In particular we prove sup u ∈ H ̃ n p, p ( Ω ), ‖ ( − Δ ) n 2 p u ‖ L p ( Ω ) ≤ 1 ∫ Ω e α n, p | u | p p − 1 d x ≤ c n, p | Ω |, for a positive constant α n, p whose sharpness we also prove. We further extend this result to the case of Lorentz-spaces (i.e. ( − Δ ) n 2 p u ∈ L ( p, q ) ). The proofs are simple, as they use Green functions for fractional Laplace operators and suitable cut-off procedures to reduce the fractional results to the sharp estimate on the Riesz potential proven by Adams and its generalization proven by Xiao and Zhai. We also discuss an application to the problem of prescribing the Q -curvature and some open problems.