TY - JOUR
T1 - Singular quasisymmetric mappings in dimensions two and greater
AU - Romney, Matthew
N1 - Publisher Copyright:
© 2019 Elsevier Inc.
PY - 2019/7/31
Y1 - 2019/7/31
N2 - 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.
AB - 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.
KW - Absolute continuity
KW - Metric space
KW - Quasiconformal mapping
UR - http://www.scopus.com/inward/record.url?scp=85065924194&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85065924194&partnerID=8YFLogxK
U2 - 10.1016/j.aim.2019.05.022
DO - 10.1016/j.aim.2019.05.022
M3 - Article
AN - SCOPUS:85065924194
SN - 0001-8708
VL - 351
SP - 479
EP - 494
JO - Advances in Mathematics
JF - Advances in Mathematics
ER -