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ABSTRACT. Suppose that f:D -• Rn is an «-dimensional J^-quasiconformal mapping. Then the first partial derivatives off are locally ZZ-integrable in D for/? e [ 1, n + c), where c is a positive constant which depends only on K and n.Suppose that D is a domain in Euclidean rc-space R n where n ^ 2, and that ƒ : D -* R n is a homeomorphism. For each xeDwe letwhere B (x, r) denotes the open ^-dimensional ball of radius r about x and m denotes Lebesgue measure in R n . We call L f (x) and J f (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 J f is locally LMntegrable there.Suppose in addition that ƒ is X-quasiconformal in D. This result is derived from the following two lemmas. The first is an inequality relating the L 1 -and L"-means of L f over small n-cubes, while AM S (MOS) subject classifications (1970). Primary 30A60; Secondary 30A86.

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Suppose that f:D-• R n is an «-dimensional J^-quasi-conformal mapping. Then the first partial derivatives off are locally ZZ-integrable in D for/? e [ 1, n + c), where c is a positive constant which depends only on K and n. Suppose that D is a domain in Euclidean rc-space R n where n ^ 2, and that ƒ : D-* R n is a homeomorphism. For each xeDwe let L/x) = lim sup | ƒ (y)-f(x)\/\y-x\, J f (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 n. We call L f (x) and J f (x\ respectively , the maximum stretching and generalized Jacobian for the homeo-morphism ƒ at the point x. These functions are nonnegative and measurable in D, and Lebesgue's theorem implies that J f is locally LMntegrable there. Suppose in addition that ƒ is X-quasiconformal in D. Then L n f ^ KJ f a.e. in D, and thus L f is locally L n-integrable in D. Bojarski has shown in [1] that a little more is true in the case where n = 2, namely that L f is locally L p-integrable in D for p e [2, 2 + c), where c is a positive constant which depends only on K. His proof consists of applying the Calderón-Zygmund 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 n and that f\D^R n isa K-quasiconformal mapping. Then L f is locally L p-integrable in D for p e [1, n + c\ where c is a positive constant which depends only on K and n. This result is derived from the following two lemmas. The first is an inequality relating the L 1-and L"-means of L f over small n-cubes, while AM S (MOS) subject classifications (1970). Primary 30A60; Secondary 30A86.