Mappings of L _p-integrable distortion: Regularity of the inverse

Let be an open set in R_n and suppose that is a Sobolev homeomorphism. We study the regularity of f _-1 under the L _p-integrability assumption on the distortion function K f. First, if is the unit ball and p > n-1, then the optimal local modulus of continuity of f _-1 is attained by a radially symmetric mapping. We show that this is not the case when p n-1 and n ≥3, and answer a question raised by S. Hencl and P. Koskela. Second, we obtain the optimal integrability results for Df _-1 in terms of the L _p-integrability assumptions of K f.