Maximal function

Abstract: 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 c 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 f\D^Risa K-quasiconformal mapping. Then Lf is locally L -integrable in D for p e [1, n + c\ where c is a positive constant which depends only on K and n.

Abstract: Weights.- Decomposition of weights.- Sharp maximal functions.- Functions in the upper half-space.- Extensions of distributions.- The Hardy spaces.- A dense class.- The atomic decomposition.- The basic inequality.- Duality.- Singular integrals and multipliers.- Complex interpolation.