What is $L^p_x L^q_t$?

Question

Often one encounters spaces written as $L^p_x L^q_t$. I was wondering what the exact definition of these spaces is. Does it just mean it is $L_p$ with respect to $x$ and $L^q$ with respect to $t$ or is there more structure?

Answer 1

Yes, that is basically what it means. To be more precise, a measurable function $$ f\colon \Omega\times I\to\mathbb{R},\,(x,t)\mapsto f(x,t) $$ is in $L^p_x L^q_t$ if for almost every $x\in \Omega$ the function $f(x,\cdot)$ is in $L^q(I)$, and the map $x\mapsto \lVert f(x,\cdot)\rVert_{L^q(I)}$ is in $L^p(\Omega)$. A different way to denote this space is $L^p(\Omega;L^q(I))$. If I remember correctly, one chapter in Evans's Partial Differential Equations treats spaces of this kind in more detail.