# TheLp-integrability of the partial derivatives of A quasiconformal mapping

@article{Gehring1973TheLpintegrabilityOT,
title={TheLp-integrability of the partial derivatives of A quasiconformal mapping},
author={Frederick W. Gehring},
journal={Acta Mathematica},
year={1973},
volume={130},
pages={265-277}
}
• 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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## References

SHOWING 1-10 OF 14 REFERENCES

### Quasiconformal mappings in space

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### On the existence of certain singular integrals

• Mathematics
• 1952
Let f (x) and K (x) be two functions integrable over the interval (-∞,+∞). It is very well known that their composition  \int\limits_{{ - \infty }}^{{ + \infty }} {f(t)K\left( {x - t} \right)dt}

### Quasi-conformal mappings inn-space and the rigidity of hyperbolic space forms

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### Symmetrization of rings in space

holds. We then estimate mod R' either by means of the space analogues of the Grötzsch and Teichmüller rings or by means of spherical annuli. The two bounds we obtain are given in Theorem 3 of §17 and

### Singular Integrals and Di?erentiability Properties of Functions

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### RINGS AND QUASICONFORMAL MAPPINGS IN SPACE.

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