# Higher Order Calderon-Zygmund Estimates for the p-Laplace Equation

## TL;DR

This paper establishes higher order regularity transfer results for the p-Laplace equation, showing that under certain conditions, the regularity of the force term F implies similar regularity for the flux A(∇u), with specific results in the planar case.

## Contribution

It extends Calderon-Zygmund estimates to higher order Besov and Triebel-Lizorkin spaces for the p-Laplace equation, including new decay estimates for p-harmonic functions in the plane.

## Key findings

- For p ≥ 2, F in B^s_{ρ,q} implies A(∇u) in B^s_{ρ,q} in the planar case.
- The result does not hold for p < 2.
- New linear decay estimates for p-harmonic functions in the plane for all p in (1, ∞).

## Abstract

The paper is concerned with higher order Calderon-Zygmund estimates for the $p$-Laplace equation $$ -\textrm{div}(A(\nabla u)) := -\textrm{div}{(|\nabla u|^{p-2}\nabla u)}=-\textrm{div} F, \qquad 1<p<\infty. $$ We are able to transfer local interior Besov and Triebel-Lizorkin regularity up to first order derivatives from the force term $F$ to the flux $A(\nabla u)$. For $p\geq 2$ we show that $F \in B^s_{\rho,q}$ implies $A(\nabla u) \in B^s_{\rho,q}$ for any $s \in (0,1)$ and all reasonable $\rho,q \in (0,\infty]$ in the planar case. The result fails for $p<2$. In case of higher dimensions and systems we have a smallness restriction on $s$. The quasi-Banach case $0<\min\{\rho,q\} < 1$ is included, since it has important applications in the adaptive finite element analysis. As an intermediate step we prove new linear decay estimates for $p$-harmonic functions in the plane for the full range $1<p<\infty$.