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 {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.

中文翻译：

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关于 Sobolev 和 Gagliardo 的注解——当 𝑝 > 𝑁 时的 Nirenberg 不等式

摘要 已知 Sobolev 空间 W 1 , p ⁢ ( ℝ N ) {W^{1,p}(\mathbb{R}^{N})} 嵌入到 LN ⁢ p / ( N - p ) ⁢ ( ℝ N ) {L^{Np/(Np)}(\mathbb{R}^{N})} 如果 p < N {p N {p>N} 。这两种不同嵌入的证明通常存在不连续性，因为对于 p > N {p>N} ，估计 ∥ 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}} 是常见的与 Hölder 范数的估计一起获得。在这篇笔记中，我们给出了 L ∞ {L^{\infty}} -embedding 的证明，它只跟随 Sobolev–Gagliardo–Nirenberg 估计的迭代 ∥ u ∥ N / ( N - 1 ) ≤ C ⁢ ∥ D ⁢ u ∥ 1 {\lVert u\rVert_{N/(N-1)}\leq C\lVert Du\rVert_{1}} . 这种证明的优点是很容易扩展到各向异性的情况，并立即导出到离散的 Lebesgue 和 Sobolev 空间的情况；我们在有限差分和有限体积方案的情况下给出样本结果。