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The aim of this paper is to establish global Calder\'{o}n--Zygmund theory toparabolic $p$-Laplacian system:  $$ u_t -\operatorname{div}(|\nabla u|^{p-2}\nabla u) = \operatorname{div}(|F|^{p-2}F)~\text{in}~\Omega\times (0,T)\subset \mathbb{R}^{n+1},  $$ proving that $$F\in L^q\Rightarrow \nabla u\in L^q,$$ for any$q>\max\{p,\frac{n(2-p)}{2}\}$ and $p>1$. Acerbi and Mingione \cite{Acerbi07}proved this estimate in the case $p>\frac{2n}{n+2}$. In this article we settlethe case $1<p\leq \frac{2n}{n+2}$. We also treat systems with discontinuouscoefficients having small BMO (bounded mean oscillation) norm.

Global Calder\'{o}n--Zygmund theory for parabolic $p$-Laplacian system: the case $1