Sobolev–Lorentz spaces in the Euclidean setting and counterexamples. (March 2017)

• Record Type: Journal Article
• Authors: Costea, Şerban
• Is Part Of: Nonlinear analysis. Volume 152(2017)
• Page Start: 149
• Page End: 182
• Publication Date: 2017-03
• Subjects: primary 31C45 46E35; Sobolev spaces -- Lorentz spaces -- Morrey embedding theorems; 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.2017.01.001
• Languages: English
• ISSNs: 0362-546X

Abstract

Abstract: This paper studies the inclusions between different Sobolev–Lorentz spaces W 1, ( p, q ) ( Ω ) defined on open sets Ω ⊂ R n, where n ≥ 1 is an integer, 1 < p < ∞ and 1 ≤ q ≤ ∞ . We prove that if 1 ≤ q < r ≤ ∞, then W 1, ( p, q ) ( Ω ) is strictly included in W 1, ( p, r ) ( Ω ) . We show that although H 1, ( p, ∞ ) ( Ω ) ⊊ W 1, ( p, ∞ ) ( Ω ) where Ω ⊂ R n is open and n ≥ 1, there exists a partial converse. Namely, we show that if a function u in W 1, ( p, ∞ ) ( Ω ), n ≥ 1 is such that u and its distributional gradient ∇ u have absolutely continuous ( p, ∞ ) -norm, then u belongs to H 1, ( p, ∞ ) ( Ω ) as well. We also extend the Morrey embedding theorem to the Sobolev–Lorentz spaces H 0 1, ( p, q ) ( Ω ) with 1 ≤ n < p < ∞ and 1 ≤ q ≤ ∞ . Namely, we prove that the Sobolev–Lorentz spaces H 0 1, ( p, q ) ( Ω ) embed into the space of Hölder continuous functions on Ω ¯ with exponent 1 − n p whenever Ω ⊂ R n is open, 1 ≤ n < p < ∞, and 1 ≤ q ≤ ∞ .