A Note on the Sobolev and Gagliardo--Nirenberg Inequality when 𝑝 > 𝑁

Abstract

Abstract It is known that the Sobolev space W 1 , p ⁢ ( ℝ N ) {W^{1,p}(\mathbb{R}^{N})} is embedded into L N ⁢ p / ( N - p ) ⁢ ( ℝ N ) {L^{Np/(N-p)}(\mathbb{R}^{N})} if p < N {p<N} and into L ∞ ⁢ ( ℝ N ) {L^{\infty}(\mathbb{R}^{N})} if p > N {p>N} . There is usually a discontinuity in the proof of those two different embeddings since, for p > N {p>N} , the estimate ∥ u ∥ ∞ ≤ C ⁢ ∥ D ⁢ u ∥ p N / p ⁢ ∥ u ∥ p 1 - N / p {\lVert u\rVert_{\infty}\leq C\lVert Du\rVert_{p}^{N/p}\lVert u\rVert_{p}^{1-N% /p}} is commonly obtained together with an estimate of the Hölder norm. In this note, we give a proof of the L ∞ {L^{\infty}} -embedding which only follows by an iteration of the Sobolev–Gagliardo–Nirenberg estimate ∥ u ∥ N / ( N - 1 ) ≤ C ⁢ ∥ D ⁢ u ∥ 1 {\lVert u\rVert_{N/(N-1)}\leq C\lVert Du\rVert_{1}} . This kind of proof has the advantage to be easily extended to anisotropic cases and immediately exported to the case of discrete Lebesgue and Sobolev spaces; we give sample results in case of finite differences and finite volumes schemes.