Pointwise Calder\'on-Zygmund gradient estimates for the $p$-Laplace system

TL;DR

This paper develops a nonlinear Calderón-Zygmund theory for the p-Laplace system, providing pointwise gradient estimates that extend classical results to nonlinear PDEs with broad norm applicability.

Contribution

It introduces a nonlinear Calderón-Zygmund framework for the p-Laplace system, enabling comprehensive gradient bounds in various function spaces.

Findings

New pointwise gradient estimates for p-Laplace solutions

Extension of Calderón-Zygmund theory to nonlinear systems

Recovery of classical results in standard function spaces

Abstract

Pointwise estimates for the gradient of solutions to the $p$-Laplace system with right-hand side in divergence form are established. They enable us to develop a nonlinear counterpart of the classical Calder\'on-Zygmund theory in terms of Calder\'on-Zygmund singular integrals, for the Laplacian. As a consequence, a flexible, comprehensive approach to gradient bounds for the $p$-Laplace system for a broad class of norms is derived. In particular, new gradient estimates are exhibited, and well-known results in customary function spaces are easily recovered.