Inequality involving Lp space, Holder Space, Sobolev Space

Question

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 $U$ is a bounded, open subset of $\mathbb{R}^{n}$ and $\partial U$ is $C^{1}$ and where $C^{0,\gamma}(\bar{U})$ is the Holder space.

Thanks a lot for any assistance! Let me know if something is unclear.

Answer 1

For the second inequality I refer you to Brezis Corollary 9.14. For the first inequality, remember that $\|u\|_\infty\leq \|u\|_{C^{0,\gamma}(\overline{U})}$, therefore $$\int_U |u|^p\leq\int_U\|u\|_\infty^p\leq |U|\|u\|_{C^{0,\gamma}(\overline{U})}^p$$

where $|U|$ is the Lebesgue measure of $U$.

Answer 2

The first inequality holds as mentioned by Tomas. The second inequality I think is a direct result of a Theorem which extends from Morrey's Inequality: If $U$ is bounded, open subset of $\mathbb{R}^{n}$, and suppose $\partial U$ is $C^{1}$. If $n < p \leq \infty$ and $u \in W^{1,p}(U)$. Then $u$ has a version $u^{*} \in C^{0,\gamma}(\bar{U})$ such that $||u^{*}||_{C^{0,\gamma}(\bar{U})} \leq C||u||_{W^{1,p}(U)}$.

Combining these two inequialites you get $||u||_{L^{p}} \leq D||u^{*}||_{C^{0,\gamma}(\bar{U})} \leq C||u||_{W^{1,p}(U)}$ where $C>0$ and $D>0$ are constants. Note that $u = u^{*}$ a.e. by definition of a version.