It suffices to show this for $l = 1$. So suppose $f$ is a locally $L_1$ function on an $n$-dimensional domain whose weak derivatives are all locally in $L_p$. By the Sobolev inequality $f$ is locally in $L_q$ for at least one $q > p$. (If $p < n$, then this holds for $q = pn/(n-p)$. If $p > n$, then it holds for $q = \infty$. If $p = n$, then it holds for all $q < \infty$.) By the Holder inequality, it follows that $f$ is also locally in $L_p$.

(Original answer edited to make it shorter)

It suffices to show this for $l = 1$. It also suffices to show that $f$ is locally in $L_q$ for some $q \ge p$. But this follows immediately by the Sobolev inequality.