Let $\Omega\_x$ and $\Omega\_y$ be sets of finite Lebesgue measure.


We can then look at the space $X\_1:=L^2(\Omega\_x \times \Omega\_y).$


This space is contained in the larger space 


$$X\_0:=L^2(\Omega\_x; L^1(\Omega\_y)).$$


On the other hand, the space $$X\_2:=L^2(\Omega\_y;L^{\infty}(\Omega\_x))$$ is smaller than $X\_1.$ 


Altogether 


$$X\_2 \subset X\_1 \subset X\_0.$$


I am wondering whether we can interpolate the canonical norm on $X\_1$ just in terms of the canonical norms on $X\_0$ and $X\_2$ such that convergence in $X\_0$ and boundedness in $X\_2$ implies convergence in $X\_1$?



Interpolation of $L^p$ spaces
No. Notice that since you assumed finiteness of the $\Omega\_x$ measure, we can restrict to looking at functions that are constant in $x$. 


Let $\Omega\_y = [0,1]$ for example, and take $\chi\_n(y) = \sqrt{n} \mathbb{1}\_{[0,1/n]}$. 


The sequence of functions $\chi\_n \overset{L^1}{\longrightarrow} 0$, and is bounded in $L^2$. But do not converge in $L^2$. 


Now extend the function so they are independent of $x$, and you have a counterexample for your triple of spaces. 


