The paper is concerned with higher order Calderón-Zygmund estimates for the p-Laplace equationWe 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 s ̺,q implies A(∇u) ∈ B s ̺,q for any s ∈ (0, 1) and all reasonable ̺, 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{̺, 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 < ∞.
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