Mappings of Lp-integrable distortion: regularity of the inverse

Let X be an open set in R-n and suppose that f : X -> R-n 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 X 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 vertical bar Df(-1)vertical bar in terms of the L-p-integrability assumptions of K-f.