*Submitted Paper*

**Inserted:** 23 may 2017

**Last Updated:** 25 may 2017

**Year:** 2017

**Doi:** https://arxiv.org/abs/1705.07438

**Abstract:**

In this paper, we prove the Lorentz space $L^{q,p}$-estimates for gradients
of very weak solutions to the linear parabolic equations with
$\mathbf{A}_q$-weights $$u_{t}-\operatorname{div}(A(x,t)\nabla
u)=\operatorname{div}(F),$$ in a bounded domain $\Omega\times
(0,T)\subset\mathbb{R}^{N+1}$, where $A$ has a small mean oscillation, and
$\Omega$ is a Lipchistz domain with a small Lipschitz constant.