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For each $t\in\mathbb{R}$, we define the Sobolev space \begin{equation} H_t=\{u\in\mathcal{S}':\int(1+|y|^2)^t|\hat{u}(y)|^2dy<+\infty\}, \end{equation} where $\mathcal{S}'$ is the space of tempered distributions on $\mathbb{R}^d$.

We give this space a norm \begin{equation} \|u\|_t=(\int(1+|y|^2)^t|\hat{u}(y)|^2dy)^{1/2}. \end{equation}

I learnt from Rudin that under these definitions the operator $k^2-\Delta$ is a homeomorphism from $H_{s+2}$ to $H_{s}$. This seems very interesting, but Rudin does not say how we can use this fact.

I wonder what can we learn from this. Any reference or suggestion is welcome.


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Note that the problem $$k^2-\Delta u=f$$ has solution for all $f\in H^s$. – Tomás Oct 30 '12 at 10:22
@Tomás Well, but that only uses the fact that $k^2-\Delta$ is surjective. – Hui Yu Oct 30 '12 at 10:27
THe solution is unique. :) – Tomás Oct 30 '12 at 10:29
up vote 1 down vote accepted

Well this problem is not that promising as I thought. I guess I will just answer it and close it.

As mentioned by Tomas. This implies the unique existence of solution to $(k^2-\Delta)u=f$.

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