Marcinkiewicz regularity for singular parabolic <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si4.svg"><mml:mi>p</mml:mi></mml:math>-Laplace type equations with measure data

We consider quasilinear parabolic equations with measurable coefficients when the right-hand side is a signed Radon measure finite total mass, having $p$-Laplace type: $$u_t - \textrm{div} \, \mathbf{a}(Du,x,t) = \mu \quad \textrm{in} \ \Omega \times (0,T) \subset \mathbb{R}^n \mathbb{R}.$$ In singular range $\frac{2n}{n+1} <p \le 2-\frac{1}{n+1}$, we establish regularity estimates for spatial gradient of solutions in Marcinkiewicz spaces, under suitable density condition measure.