The Bourgain space is $X^{s,b} := X^{s,b}(\mathbb R \times \mathbb{T}^3)$ is the completion of $C^\infty (\mathbb R; H^s(\mathbb{T}^3))$ under the norm
$$\| u\|_{X^{s,b}}:= \|e^{- i t \triangle u(t,x)}\|_{H_t^b (\mathbb{R}; H_x^s(\mathbb{T}^3))}\\ =\left(\sum_{\xi \in \mathbb{Z}^3} \int_\mathbb{R} <\tau+\sum_{j=1}^3 \theta_j \xi^2_j>^{2b} <\xi>^{2s} |\hat{u}(\tau,\xi)|^2 d\tau \right)^{\frac{1}{2}}.$$
I am trying to prove the following:
1- For $s_1 \leq s_2$ and $b_1 \leq b_2$, $X^{s_2,b_2} \hookrightarrow X^{s_1,b_2}$.
2- For $b > 1/2$, $X^{0,b} \hookrightarrow C_t L_x^2$.
3- $X^{0,1/4} \hookrightarrow L_t^4 L_x^2$
I am new to mixed norms. However, I solved the first point by, as in the comments, taking the bounded remaining term out. I am confused with the second and third points? Any help.