# Tagged Questions

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### Relation between Sobolev Space $W^{1,\infty}$ and the Lipschitz class

I have a Sobolev space related question. In the book 'Measure theory and fine properties of functions' by Lawrence Evans. I know the result that states that for $f: \Omega \rightarrow \mathbb{R}$. $f$ ...
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### Hölder-Zygmund Spaces on compact sets and for integer smoothness parameters

I know from Triebl, Theory of Function Spaces II, that for $\alpha \notin \mathbb{N}$ Hölder-Zygmund Spaces on $\mathbb{R}$ are equal to the classical Hölder Spaces. However, I have two questions ...
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### Product of Hölder and Sobolev functions

Here $C^{\kappa , \lambda} ( \overline{\Omega} ) = \left\{ h|_{\overline{\Omega}} :h \in C^{\kappa , \lambda} ( \mathbb{R}^{n} ) \text{ and } h \text{ has compact support} \right\}$ denotes $\kappa$ ...
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### Sobolev, Holder, Lp spaces continuous and compact embeddings proof

I would like to know if the following proof is fine. I haven't filled in all the detail but please let me know what you think about the basic outline.(I am aware that there are posts which have dealt ...
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### Showing $u \in H^s$ and $\varphi \in C^1$ implies $u\varphi \in H^s$ (product rule)

Let $\Omega$ be bounded and open set in $\mathbb{R}^n$. As a start, I pose this question: For $u \in H^s(\Omega)=W^{s,2}(\Omega)$, define the Holder seminorm type quantity F(u) = ...
Do we have the following inequality or some variation of this: $||u||_{L^{p}(U)} \leq ||u||_{C^{0,\gamma}(\bar{U})} \leq ||u||_{W^{1,p}(U)}$ for $n < p \leq \infty$ for $u \in W^{1,p}(U)$ where $U$ ...