Consider $\Omega \subset \mathbb{R}^n$ open and bounded, $I$ some bounded interval of $\mathbb{R}$. Let $2^\ast := \frac{2n}{n-2}$ (the critical Sobolev exponent).
Let $u \in L^{\infty}(I; L^2(\Omega)) \cap L^2(I; L^{2^\ast}(\Omega))$. Let $(p,q)$ such that $$\frac{2}{p} + \frac{n}{q} = \frac{n}{2}.$$ How can I control the norm of $u$ in $L^p(I; L^q(\Omega))$ with the norms in $L^{\infty}(I; L^2(\Omega))$ and $L^2(I; L^{2^\ast}(\Omega))$?
I am aware of another question in this direction but the relation between $p$ and $q$ is not the same here.
This appears in these notes, at the bottom of page 17.
Actually, the post you linked basically answers your question. You simply need to rewrite $$ \frac{1}{p}= \frac{(1-\theta)}{\infty}+ \frac{\theta}{2},\quad \frac{ 1}{q}= \frac{(1-\theta)}{2}+ \frac{\theta}{2^*}$$ with the same number $\theta\in[0,1]$. This can be done since the pairs $(p_1,q_1)=(\infty,2)$ and $(p_2,q_2)=(2,2*)$ satisfy the same relation of $(p,q)$, that is, $$ \frac{ 2}{p_j}+ \frac{ n}{q_j}= \frac{ n}{2}, $$ $j=1,2$. Once you do this, you simply apply the arguments in the linked post and you are done.