I have difficulty figuring out two norms are equivalent.

Let s be a nonnegatiev real number and K is a compact subset of $\mathbb{R}^d$. Let $H_{K}^s(\mathbb{R}^d)$ be the space of distributions of $H^s(\mathbb{R}^d)$ which are supported in K. Then for every $u\in H_{K}^s(\mathbb{R}^d)$, $||u||_{H^s}$ and $||u||_{\dot{H^S}}$ are equivalent.

In the book it says we only need to show $||u||_{L^2}\leq C||u||_{\dot{H^s}}$. I know how to show this, but I don't why by showing this equality we can get the answer. I only know u is compactly supported, what about $\hat{u}$? I guess probably it is not compactly supported. Then I don't know what to do. Thanks for any hint!

  • $\begingroup$ How is $\dot H^s_K$ and its norm defined? And what is this $\hat u$ you're talking about? (And what book are you referring to?) $\endgroup$ – Three.OneFour Nov 4 '15 at 17:02

Two norms $\|\cdot\|_1$ and $\|\cdot\|_2$ are equivalent iff there is a constans $\alpha$ and $\beta$ such that for all $u$ $$\alpha \|u\|_1\leq \|u\|_2\leq\beta \|u\|_1$$ I guess the book asking only one side because the other is already discussed or trivial or so.

  • $\begingroup$ Yes. One side is trivial. And the book shows the other side by showing L2 norm is smaller than $\dot{H^s}$ norm. I know why L2 is smaller than $ \dot{H^s}$ norm, but I don't know why it gives us the answer. $\endgroup$ – cali Nov 4 '15 at 16:47
  • $\begingroup$ Are you asking why this is the definition of the equivalency? Look here math.stackexchange.com/questions/1380191/… $\endgroup$ – Michael Medvinsky Nov 4 '15 at 16:51
  • $\begingroup$ No. The L2 is not the same as the inhomogenous sobolev norm, but the book says we only need to show the L2 norm. This is where I got confused. $\endgroup$ – cali Nov 4 '15 at 17:03
  • $\begingroup$ If $\alpha \|u\|_{H_s}\leq \|u\|_{L_2}$ is ok with you and the book shows $\|u\|_{L_2}\leq\beta \|u\|_{H_s}$ then you should be convinced that they are equivalent. But I guess I didn't get your problem yet. $\endgroup$ – Michael Medvinsky Nov 4 '15 at 17:13
  • $\begingroup$ Why $||u||_{H^s}$ is comparable to $||u||_{L^2}$? I guess that's my question. $\endgroup$ – cali Nov 4 '15 at 17:26

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