$L^1(0,T;(L^1_{loc}(\mathbb{R^N}))^N)$ - Confused about notations used for space and time dependent vector fields

Question

I found this notation - $L^1(0,T;(L^1_{loc}(\mathbb{R^N}))^N)$ - in a paper of DiPerna and Lions concerning vector fields space and time dependent, "Ordinary differential equations, transport theory and Sobolev spaces" - Invent. math. 98, 511-547 (1989), and I wasn't able to understand which function they are actually considering. I tried to search on the net but I wasn't able to find it stated. Supposing this notation to be "standard" in the field could someone please help me with it?

Answer 1

A function $u$ is in this space if $$\int_0^T\|u(t)\|_{L^1(K;\mathbb R^N)}<\infty,$$ for all $K\subset\mathbb R^N$ that are compact.