In this paper we study the regularity of the so-called very weak solution for a parabolic equation. This unique solution is only integrable over the parabolic cylinder. The initial data and the right-hand side of the linear parabolic equation are functions integrable with respect to the weight function which corresponds to the distance function. In particular, we prove some global regularity of the space-gradient in Lorentz spaces. The regularity with respect to the time derivative is obtained under the condition that the linear operator is time independent and self-adjoint via $m$-accretive theory.
"Regularity of a very weak solution for parabolic equations and applications." Adv. Differential Equations 16 (9/10) 867 - 894, September/October 2011.