Thinking about the partial derivative in this question $\Delta u$ is bounded. Can we say $u\in C^1$? of mine, I encountered this post.
Equivalent Norms on Sobolev Spaces
I wonder if this hold when $\alpha\neq 2$ as well.
Too many details are omitted in this post. Could I ask a reference?
Yes, it is true.
This is actually an general idea for space involving several order of derivatives: "the extreme terms in a sum often already suffice to control the intermediate terms". Notice that by extreme we mean both highest order and the lowest order.
For example, $W^{3,p}$ norm of $u$ can be controlled by using only $L^p$ norm of $u$ and the $L^p$ norm of 3rd derivative of $u$.
This idea also applied on space $C^p(\Omega)$, the continuous differentiable function space of order $p$ with $L^\infty$ norm. Also, Holder space is applied as well.
For a good reference of this idea, I would suggest you to read this post by Terence Tao, look for exercise 2 for more explanation.
Also, for Equivalent Norms on Sobolev spaces, first look at this post, Theorem 2.7 for a summarization, look this book, page 133, theorem 5.2 for details proof. (The proof is not short)
