Gradient weighted estimates at the natural exponent for Quasilinear Parabolic equations

In this paper, we obtain weighted norm inequalities for the spatial gradients of weak solutions to quasilinear parabolic equations with weights in the Muckenhoupt class $A_{\frac{q}{p}}(\mathbb{R}^{n+1})$ for $q\geq p$ on non-smooth domains. Here the quasilinear nonlinearity is modelled after the standard $p$-Laplacian operator. Until now, all the weighted estimates for the gradient were obtained only for exponents $q>p$. The results for exponents $q>p$ used the full complicated machinery of the Calder\'on-Zygmund theory developed over the past few decades, but the constants blow up as $q \rightarrow p$ (essentially because the Maximal function is not bounded on $L^1$). In order to prove the weighted estimates for the gradient at the natural exponent, i.e., $q=p$, we need to obtain improved a priori estimates below the natural exponent. To this end, we develop the technique of Lipschitz truncation based on \cite{AdiByun2,KL} and obtain significantly improved estimates below the natural exponent. Along the way, we also obtain improved, unweighted Calder\'on-Zygmund type estimates below the natural exponent which is new even for the linear equations.