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The space is $H^s(\mathbb R^d)$. If $f$ is in this space, it means

$\int_\mathbb {R^n} (1+|\xi|^2)^s|\hat f(\xi)|^2d\xi < \infty$

where $\hat f$ is the fourier transform of $f$: $\hat f(\xi)=\int_\mathbb {R^n} f(x)e^{-2\pi ix\cdot \xi}dx$.

Any good properties for this space?

I found out that if s=1, than $f$, $\nabla f\in L^2$

if s=2, then $f,\nabla f, \Delta^2f\in L^2$.

My goal is to prove

$f(x)=\frac{1}{2\pi}\int \hat f(\xi) e^{i(\xi,x)}d\xi$ where$(\xi, x)$ is the inner product=$x_1\xi_1 +x_2\xi_2...$

And I want to show that if $f\in H^s$, $|f|_{L^\infty}\leq c|f|_{H^2}$

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    $H^s$ is a *Sobolev space*, and the statement you want to prove is the *Fourier inversion formula*. Both of these will be discussed at length in most graduate real analysis textbooks. Can you be more specific about what kind of "good properties" you want?2012-12-14
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    I have heard that Hilbert walked into Courant's office one day and asked: “What is this thing they call Hilbert space?”. I have no idea if the story is true, but the title of your question reminded me of it. Anyhow, the Laplace operator is usually written as $\Delta$ or $\nabla^2$. What you wrote, $\Delta^2$, would be the bi-Laplacian. You'd need to be in $H^4$ to ensure that $\Delta^2f\in L^2$.2012-12-14
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    But I thought the incersion formula is kind of different from this one.2012-12-14

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This is a Sobolev space. These spaces have many nice properties! The link there (and here) should get you going...