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

TL;DR

This paper establishes Lorentz space gradient estimates for very weak solutions of linear parabolic equations with BMO coefficients, under weighted conditions and small oscillation assumptions, in Lipschitz domains.

Contribution

It provides new Lorentz space $L^{q,p}$-estimates for gradients of very weak solutions to parabolic equations with BMO coefficients, extending previous regularity results.

Findings

Lorentz space estimates for gradients of solutions

Results hold for equations with small mean oscillation coefficients

Applicable in Lipschitz domains with small Lipschitz constant

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.