The Lp-integrability of the partial derivatives of A quasiconformal mapping

Jf (x) = lim sup m (f (B (x, r) ) ) /m (B (x, r) ), r->0 where B (x, r) denotes the open ^-dimensional ball of radius r about x and m denotes Lebesgue measure in R. We call Lf (x) and Jf (x respectively, the maximum stretching and generalized Jacobian for the homeomorphism ƒ at the point x. These functions are nonnegative and measurable in D, and Lebesgue's theorem implies that Jf is locally LMntegrable there. Suppose in addition that ƒ is X-quasiconformal in D. Then Lf ^ KJf a. e. in D, and thus Lf is locally L -integrable in D. Bojarski has shown in 1 that a little more is true in the case where n = 2, namely that Lf is locally L-integrable in D for p e 2, 2 + c), where is a positive constant which depends only on K. His proof consists of applying the CalderonZygmund inequality [2 to the Hubert transform which relates the complex derivatives of a normalized plane quasiconformal mapping. Unfortunately this elegant two-dimensional argument does not suggest what the situation is when n > 2. The purpose of this note is to announce the following n-dimensional version of Bojarski's theorem. THEOREM. Suppose that D is a domain in R and that fRisa K-quasiconformal mapping. Then Lf is locally L -integrable in D for p e [1, n + c where is a positive constant which depends only on K and n.