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

Let _(x, !)r |!| p&2 ! be a p-Laplacian type operator and consider the Hodge decomposition _(x, Du)=D.+H, div H=0. A standard elliptic theory asserts that &D.&q (p&1) C &Du& p&1 q for all q>p&1. There has been considerable recent interest in the validity of the reverse estimate &Du& p&1 q 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. 2001 Academic Press