Let $T$ be a distribution on $\mathbb{R}^{n}$ such that there are functions $f_1,\ldots,f_n \in L^1_{loc}(\mathbb{R}^n)$ so that $\dfrac{\partial T}{\partial x_{j}} = f_j, \forall j=1,...,n. $
My question: How to prove that $T$ is indeed a function in $L^1_{loc}(\mathbb{R}^n)$, i.e. there is $u\in L^1_{loc}(\mathbb{R}^n$) such that $$T(\phi) = \int_{\mathbb{R}^n}u \phi , \forall \phi\in C_{0}^{\infty}(\mathbb{R}^n)?$$
When $n=1$, a proof can be found here. But I could not adapt this since I am not sure how to use the absolute continuity mentioned in one dimension to the case $n>1$.
