Gradient weighted estimates at the natural exponent for quasilinear parabolic equations

• Published in: Advances in mathematics (New York. 1965) 2019-05, Vol.348, p.456-511
• Main Authors: Adimurthi, Karthik, Byun, Sun-Sig
• Format: Article
• Language: English
• Subjects: Lipschitz truncation; Muckenhoupt weights; Quasilinear parabolic equations
• ISSN: 0001-8708; 1090-2082
• DOI: 10.1016/j.aim.2019.03.015

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 Aqp(Rn+1) for q≥p on non-smooth domains. Here the quasilinear nonlinearity is modeled 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 Calderón-Zygmund theory developed over the past few decades, but the constants blow up as q→p (essentially because the Maximal function is not bounded on L1). 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 [3,25] and obtain significantly improved estimates below the natural exponent. Along the way, we also obtain improved, unweighted Calderón-Zygmund type estimates below the natural exponent which is new even for the linear equations.