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 $f\in H^s$, $|f|_{L^\infty}\leq c|f|_{H^2}$