TheLp-integrability of the partial derivatives of A quasiconformal mapping

  title={TheLp-integrability of the partial derivatives of A quasiconformal mapping},
  author={Frederick W. Gehring},
  journal={Acta Mathematica},
  • F. Gehring
  • Published 1 July 1973
  • Mathematics
  • Acta Mathematica
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… 

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  • F. Gehring
  • Mathematics
    Proceedings of the National Academy of Sciences of the United States of America
  • 1961