Higher order Calderón-Zygmund estimates for the p-Laplace equation

Abstract The paper is concerned with higher order Calderon-Zygmund estimates for the p-Laplace equation − div ( A ( ∇ u ) ) : = − div ( | ∇ u | p − 2 ∇ u ) = − div F , 1 p ∞ . 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 ( ∇ u ) . For p ≥ 2 we show that F ∈ B ϱ , q s implies A ( ∇ u ) ∈ B ϱ , q s for any s ∈ ( 0 , 1 ) and all reasonable ϱ , q ∈ ( 0 , ∞ ] 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 ⁡ { ϱ , 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 ∞ .