Embeddings for \(W^{1,p}\) Link to heading
\(p< n\) (Gagliardo-Nirenberg-Sobolev type inequalities) Link to heading
- \(W^{1,p}(\mathbb{R}^n)\cap C^1_{c}(\mathbb{R}^n)\hookrightarrow L^{p^*}(\mathbb{R}^n)\)
- \(W^{1,p}(U)\hookrightarrow L^{p^*}(U)\) with bounded \(U\subset \mathbb{R}^n\) and \(\partial U\) is \(C^1\)
- \(W^{1,p}(U)\subset \subset L^{q}(U)\) with bounded \(U\subset \mathbb{R}^n\) and \(\partial U\) is \(C^1\), where \(q< p^*\)
- \(W^{1,p}_0(U)\hookrightarrow L^{p^*}(U)\) with bounded \(U\subset \mathbb{R}^n\)
\(p>n\) (Morray type inequalities) Link to heading
- \(W^{1,p}(\mathbb{R}^n)\cap C^1(\mathbb{R}^n)\hookrightarrow C^{0,\gamma}(\mathbb{R}^n)\)
- \(W^{1,p}(U)\hookrightarrow C^{0,\gamma}(U)\) with bounded \(U\subset \mathbb{R}^n\) and \(\partial U\) is \(C^1\)