**Definition** For $f\in\mathcal{S}'(\mathbb{R}^n)$, we say that $f$ belongs to the Sobolev space $H^s(\mathbb{R}^n)$ if and only if $$\norm{f}_{H^s} = \norm{ (1+\abs{\xi}^2)^{s/2} \hat u(\xi) }_{L^2} < \infty.$$ === Interpolation between the Sobolev spaces For any $s_1<s<s_2$, we know that there exists $\theta \in (0,1)$ such that $$s=\theta s_1 + (1-\theta)s_2,$$ then we have $$\norm{f}_{H^s} \leq \norm{f}_{H^{s_1}}^\theta \norm{f}_{H^{s_2}}^{1-\theta}.$$ **//Proof//** By definition, we have $$\begin{split} \norm{f}_{H^s}^2 &= \int (1+\abs{\xi}^2)^{s} \hat u(\xi)^2 d\xi ~\\ &=\int ((1+\abs{\xi}^2)^{s_1}\hat u(\xi)^2)^\theta ((1+\abs{\xi}^2)^{s_2}\hat u(\xi)^2)^{1-\theta} d\xi ~\\ &\leq (\int (1+\abs{\xi}^2)^{s_1} \hat u(\xi)^2 d\xi)^\theta (\int (1+\abs{\xi}^2)^{s_2} \hat u(\xi)^2 d\xi)^{1-\theta}~\\ &=\norm{f}_{H^{s_1}}^{2\theta} \norm{f}_{H^{s_2}}^{2(1-\theta)}, \end{split}$$ where the last inequality used H$\ddot{\text{o}}$lder's inequality with $p=1/\theta,q=1/(1-\theta)$.