# 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} }

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

- Mathematics
- 1963

U' denotes the image of U, the disk | s — So| <r, and m denotes Lebesgue plane measure. If w(z) is difîerentiable a t So, then w(z) is locally affine at z<> and maps the infinitesimal circles | z —…

### The $L^p$-integrability of the partial derivatives of a quasiconformal mapping

- Mathematics
- 1973

. 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…

### 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

- Mathematics
- 1968

© Publications mathématiques de l’I.H.É.S., 1968, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http://…

### Symmetrization of rings in space

- Mathematics
- 1961

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

- Physics
- 1971

A plurality of disks include solid laserable material and are in parallel spaced apart relation within a transparent tubular enclosure. Spaces between the disks constitute portions of a collant fluid…

### RINGS AND QUASICONFORMAL MAPPINGS IN SPACE.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 1961