Total list weighting of graphs with bounded maximum average degree

A proper total weighting of a graph $G$ is a mapping $?$ which assigns to each vertex and each edge of $G$ a real number as its weight so that for any edge $uv$ of $G$, ${\sum }_{e\in E\left(v\right)}?\left(e\right)+?\left(v\right)\ne {\sum }_{e\in E\left(u\right)}?\left(e\right)+?\left(u\right)$. A $(k,k^{\prime })$-list assignment of $G$ is a mapping $L$ which assigns to each vertex $v$ a set $L\left(v\right)$ of $k$ permissible weights and to each edge $e$ a set $L\left(e\right)$ of $k^{\prime }$ permissible weights. An $L$-total weighting is a total weighting $?$ with $?\left(z\right)\in L\left(z\right)$ for each $z\in V\left(G\right)\cup E\left(G\right)$. A graph $G$ is called $(k,k^{\prime })$-choosable if for every $(k,k^{\prime })$-list assignment $L$ of $G$, there exists a proper $L$-total weighting. It was proved in Tang and Zhu (2017) that if $p\in \{5,7,11\}$, a graph $G$ without isolated edges and with $\mathrm{mad}\left(G\right)\le p?1$ is $(1,p)$-choosable. In this paper, we strengthen this result by showing that for any prime $p$, a graph $G$ without isolated edges and with $\mathrm{mad}\left(G\right)\le p?1$ is $(1,p)$-choosable.