Equivalent norm for the Sobolev space $H^{\frac{1}{2}}(\mathbb{R}^3)$

Question

Consider the Sobolev space $H^{\frac{1}{2}}(\mathbb{R}^3)$. If $u\in H^{\frac{1}{2}}(\mathbb{R}^3)$, it is true that an equaivalent norm of $H^{\frac{1}{2}}(\mathbb{R}^3)$ is given by $$\Vert (1-\Delta)^{\frac{1}{4}}u\Vert_{L^2(\mathbb{R}^3)}?$$

If it is true, could anyone explain me why or give some references?

Thank you in advance!

Answer 1

Yes, it's true. In case you find the notation confusing, it's using the Fourier multiplier functional calculus. In particular, your norm translates to $$\|\mathcal{F}^{-1} (\langle\xi\rangle^{1/2}\mathcal{F}u)\|_{L^2(\mathbb{R}^3)}.$$ In fact, it is usually shown (using basic properties of the Fourier transform) that this definition is equivalent to the usual one for non-negative integer-order Sobolev spaces, then it is taken as the definition for real numbers.

See chapter 4 of Michael Taylor's Partial Differential Equations I: Basic Theory.