Singular quasisymmetric mappings in dimensions two and greater
Romney, M. (2019). Singular quasisymmetric mappings in dimensions two and greater. Advances in Mathematics, 31, 479-494. https://doi.org/10.1016/j.aim.2019.05.022
Published inAdvances in Mathematics
© 2019 Elsevier Inc.
For all n ≥2, we construct a metric space (X, d)and a quasisymmetric mapping f:[0, 1]n→X with the property that f−1 is not absolutely continuous with respect to the Hausdorff n-measure on X. That is, there exists a Borel set E⊂[0, 1]n with Lebesgue measure |E| >0 such that f(E) has Hausdorff n-measure zero. The construction may be carried out so that X has finite Hausdorff n-measure and |E| is arbitrarily close to 1, or so that |E| =1. This gives a negative answer to a question of Heinonen and Semmes.
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Related funder(s)Academy of Finland; European Commission
Funding program(s)Research post as Academy Research Fellow, AoF
The content of the publication reflects only the author’s view. The funder is not responsible for any use that may be made of the information it contains.
Additional information about fundingThis research was supported by the Academy of Finland grant 288501 and by the ERC Starting Grant 713998 GeoMeG.
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