In the proof of a compactness theorem involving fractional derivatives in Temam's Navier-Stokes Equations, an argument as the following is made.

Suppose $X_0,X,X_1$ are Hilbert spaces such that $ X_0\overset{\textrm{cpt}}{\hookrightarrow} X\hookrightarrow X_1. $ Suppose $\gamma>0$. Suppose $v_m$ is a sequence all supported in a bounded subset $K$ of $\mathbb{R}$ such that

• $v_m\to 0$ in $L^2(\mathbb{R};X_0)$ weakly; (2.29)
• $|\xi|^\gamma\hat{v_m}\to 0$ in $L^2(\mathbb{R};X_1)$ weakly. (2.30)

Then for $M>0$, $$ \lim_{m\to\infty}\int_{|\xi|\leq M}\|\hat{v_m}(\xi)\|_{X_1}^2\ d\xi\to 0. $$

Would anybody elaborate why this is true?