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

It is known that the Sobolev space {W^{1,p}(\mathbb{R}^{N})} is embedded into {L^{Np/(N-p)}(\mathbb{R}^{N})} if {p<N} and into {L^{\infty}(\mathbb{R}^{N})} if {p>N}. There is usually a discontinuity in the proof of those two different embeddings since, for {p>N}, the estimate {\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^{\infty}}-embedding which only follows by an iteration of the Sobolev–Gagliardo–Nirenberg estimate {\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.