Fourier Sobolev space on Euclidean space

For real $s$ , the Fourier Sobolev space $H^s(\mathbb R^n)$ consists of tempered distributions $u$ whose Fourier transform is represented by a measurable function and satisfies

\|u\|_{H^s}^2=\int_{\mathbb R^n}(1+4\pi^2|\xi|^2)^s|\widehat u(\xi)|^2\,d\xi<\infty.

The Fourier convention is $\widehat f(\xi)=\int e^{-2\pi ix\cdot\xi}f(x)\,dx$ . Negative $s$ are allowed; their weights decay at high frequency.

Integer orders and duality

Plancherel's theorem and $\widehat{\partial_j u}=2\pi i\xi_j\widehat u$ identify $H^k$ , for nonnegative integers $k$ , with $W^{k,2}(\mathbb R^n)$ with equivalent norms. Replacing $1+4\pi^2|\xi|^2$ by $1+|\xi|^2$ also gives an equivalent norm. The pairing extending the $L^2$ inner product identifies $H^{-s}$ with the continuous dual of $H^s$ . Cauchy–Schwarz in Fourier space gives $|\langle u,v\rangle|\le\|u\|_{H^{-s}}\|v\|_{H^s}$ .

Low-frequency condition

These are inhomogeneous spaces: the weight stays positive near $\xi=0$ . A delta distribution in frequency is not a weighted $L^2$ function. In particular, a nonzero constant function on the whole space belongs to no $H^s(\mathbb R^n)$ , even when $s<0$ .