On a Reverse Estimate for Hodge Decompositions of p-Laplacian Type Operators

Let σ(x, ξ)≈ξp−2 ξ be a p-Laplacian type operator and consider the Hodge decomposition σ(x, Du)=D +H, div H=0. A standard elliptic theory asserts that D q/(p−1) C Du p−1q for all q>p−1. There has been considerable recent interest in the validity of the reverse estimate Du p−1q C D q/(p−1) for q>p−1 in the regularity study of certain geometrical mappings. In this paper, we give a relatively new proof of a well-known theorem that this reverse estimate holds for all q sufficiently close to the natural power p and also prove that the estimate holds for all q p−1 for certain special weak solutions u.