Some remarks about metric spaces, spherical mappings, functions and their derivatives

@article{Semmes1996SomeRA,
  title={Some remarks about metric spaces, spherical mappings, functions and their derivatives},
  author={S. Semmes},
  journal={Publicacions Matematiques},
  year={1996},
  volume={40},
  pages={411-430}
}
  • S. Semmes
  • Published 1 July 1996
  • Mathematics
  • Publicacions Matematiques
If $p \in {\bold R}^n$, then we have the radial projection map from ${\bold R}^n \backslash \{p\}$ onto a sphere. Sometimes one can construct similar mappings on metric spaces even when the space is nontrivially different from Euclidean space, so that the existence of such a mapping becomes a sign of approximately Euclidean geometry. The existence of such spherical mappings can be used to derive estimates for the values of a function in terms of its gradient, which can then be used to derive… 

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References

SHOWING 1-10 OF 13 REFERENCES

Finding curves on general spaces through quantitative topology, with applications to Sobolev and Poincaré inequalities

In many metric spaces one can connect an arbitrary pair of points with a curve of finite length, but in Euclidean spaces one can connect a pair of points with a lot of rectifiable curves, curves that

Good metric spaces without good parameterizations

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On the nonexistence of bilipschitz parameterizations and geometric problems about $A_\infty$-weights

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TheLp-integrability of the partial derivatives of A quasiconformal mapping

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

Bilipschitz Mappings and Strong a 1 Weights

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Stein

Stein (Continued from page 4) preting what the eye sees in perspective. Image Of Behavior Are images deceptive? Are they apt to change the perceived character of the original by changing his (or its)

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