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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

  1. I wonder if this hold when $\alpha\neq 2$ as well.

  2. Too many details are omitted in this post. Could I ask a reference?

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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)

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  • $\begingroup$ Just adding Theorem 5.2 in Adams--Fournier 2nd ed is on p. 135 at least in the one I have. $\endgroup$ – shall.i.am Sep 7 '15 at 1:08

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