LAUSR

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.