Gradient weighted norm inequalities for very weak solutions of linear parabolic equations with BMO coefficients

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 $$u_t-\mathrm{div}(A(x,t)\mathrm{\nabla }u)=\mathrm{div}(F),$$ in a bounded domain $\Omega\times (0,T)\subset$\mathbb{R}$^{N+1}$,where$A$hasasmallmeanoscillation,and$\Omega$ is a Lipchistz domain with a small Lipschitz constant.