# Sobolev Embedding

#### lvleph

This question could also go in Differential Equations, but I felt it would be more likely answered here. I just need a small question answered about a norm.

What is the norm $$\displaystyle \|\cdot\|_{C^m(\Omega)}$$ in the following:
For $$\displaystyle 0\le m < k - \frac{n}{p}$$
$$\displaystyle W^{k,p}_0(\Omega) \subset C^m(\bar{\Omega}),$$
i.e., $$\displaystyle \|u\|_{C^m(\Omega)} \le c\|u\|_{W^{k,p}_0}.$$

#### Jose27

MHF Hall of Honor
This question could also go in Differential Equations, but I felt it would be more likely answered here. I just need a small question answered about a norm.

What is the norm $$\displaystyle \|\cdot\|_{C^m(\Omega)}$$ in the following:
For $$\displaystyle 0\le m < k - \frac{n}{p}$$
$$\displaystyle W^{k,p}_0(\Omega) \subset C^m(\bar{\Omega}),$$
i.e., $$\displaystyle \|u\|_{C^m(\Omega)} \le c\|u\|_{W^{k,p}_0}.$$
It's the usual, ie. $$\displaystyle \| u\| _{C^m(\Omega )} = \sum_{|\alpha | \leq m} \| D^{\alpha } u \| _{\infty } = \sum_{ |\alpha |\leq m } \sup_{\Omega } |D^{\alpha }u |$$.

• lvleph

#### lvleph

That is what I thought.