When dealing with time-dependent PDEs, one often obtain that some quantity $E(t,x)$ belongs to a Lebesgue space $L^p_t(L^q_y)$, which means that $$\int_0^{+\infty}\|E(t,\cdot)\|_{L^q(\mathbb{R}^n)}^p dt<+\infty.$$ Sometimes, we even have $E\in L^{p_1}_t(L^{q_1}_y)\cap L^{p_2}_t(L^{q_2}_y)$ ; when this occurs, a simple interpolation, which involves only Hölder inequality, yields $E\in L^p_t(L^q_y)$ where $(\frac1p,\frac1q)$ is any point in the segment between $(\frac1{p_1},\frac1{q_1})$ and $(\frac1{p_2},\frac1{q_2})$.

I am interested in a slightly different situation, where $E\in L^{p_1}_t(L^{q_1}_y)\cap L^{q_2}_y(L^{p_2}_t)$ (the order of integrations differ in both spaces). What are the interpolation spaces ?

For instance, suppose that $E\in L^\infty_t(L^1_y)\cap L^\infty_y(L^1_t)$. What can be said of $E$ ?