I need the embedding, for $I\subset\mathbb{R}$ is a bounded intervall and $\Omega\subset\mathbb{R}^n$ is a bounded domain,
$$L^{q_1}(I;L^{p_1}(\Omega))\cap L^{q_2}(I;L^{p_2}(\Omega))\hookrightarrow L^q(I;L^p(\Omega))$$
I know for $f \in L^{p_1}(\Omega) \cap L^{p_2}(\Omega)$ and $1\leq q_2\leq q\leq q_1<\infty$, it holds $f \in L^q(\Omega)$ and it holds the interpolation inequality
$$|f|_p \leq |f|_{p_1}^{1-\theta}|f|_{p_2}^\theta$$
with $\tfrac{1}{p} =: \tfrac{1-\theta}{p_1} + \tfrac{\theta}{p_2}$ or $\theta:= \tfrac{p_2}{p}\tfrac{p_1-p}{p_1-p_2}$ for $p_2 \neq p_1$.
This holds two times but how can I combine those to get my claim?
I think I need something like (for $I=(0,T)$)
$$||f||_{L^q(0,T;L^p(\Omega))}\leq||f||^{1-\theta}_{L^{q_1}(0,T;L^{p_1}(\Omega))}||f||^{\theta}_{L^{q_2}(0,T;L^{p_2}(\Omega))}$$
So I started with
$$||f||^{}_{L^q(0,T;L^p(\Omega))}=||(||f||_{L^p(\Omega)})||_{L^q(0,T)}\leq||(||f||^{1-\theta}_{L^{p_1}(\Omega)}||f||^{\theta}_{L^{p_2}(\Omega)})||^{}_{L^{q}(0,T)}$$
$$\leq||(||f||^{1-\theta}_{L^{p_1}(\Omega)})||^{}_{L^{q_1}(0,T)}\cdot||(||f||^{\theta}_{L^{p_2}(\Omega)})||^{}_{L^{q_2}(0,T)}.\tag{1}\label{1}$$
Now the big question is: Does the following inequality hold?
$$\eqref{1}\leq||f||^{1-\theta}_{L^{q_1}(0,T;L^{p_1}(\Omega))}||f||^{\theta}_{L^{q_2}(0,T;L^{p_2}(\Omega))}$$