Global Calder{o}n--Zygmund theory for parabolic $p$-Laplacian system: the case $1<pleq frac{2n}{n+2}$

The aim of this paper is to establish global Calder{o}n--Zygmund theory to parabolic $p$-Laplacian system: $$ u_t -operatorname{div}(| abla u|^{p-2} abla u) = operatorname{div} (|F|^{p-2}F)~text{in}~Omegatimes (0,T)subset mathbb{R}^{n+1}, $$ proving that $$Fin L^qRightarrow abla uin L^q,$$ 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 settle the case $1<pleq frac{2n}{n+2}$. We also treat systems with discontinuous coefficients having small BMO (bounded mean oscillation) norm.