Sobolev–Lorentz spaces in the Euclidean setting and counterexamples. (March 2017) - Costea, Şerban
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 ≤ ∞ .