I'm wondering: What is the tensor product of $L^p({\bf R})$ with $L^q({\bf R})$?

(For p=q=2, the answer clearly should be $L^2({\bf R}^2)$; for other values of $p$ and $q$, it is not at all obvious to me what $L^p({\bf R})\otimes L^q({\bf R})$ should be.)