Sharp Examples for Planar Quasiconformal Distortion of Hausdorff Measures and Removability
Mathematics Department, 202 Mathematical Sciences Bldg., University of Missouri, Columbia, MO 65211-4100, USA and Fields Institute, 222 College Street, Toronto, Ontario M5T 3J1, Canada
Correspondence: Correspondence to be sent to: ignacio{at}math.missouri.edu
In the celebrated paper [5, Acta Mathematica, 173, 1994], Astala showed optimal area distortion bounds and dimension distortion estimates for planar quasiconformal mappings. He asked (Question 4.4) whether a finer result held, namely absolute continuity of Hausdorff measures under push-forward by quasiconformal mappings. This was proved in one particular case relevant for removability questions, in the joint work of Astala, Clop, Mateu, Orobitg and the author [6, Duke Mathematical Journal, forthcoming] (Theorem 1.1), the other cases remaining open. A related question that we left open in [6] (Question 4.2) (which was asked by Astala to the author before [6] in an equivalent form [4, Personal communication]) is whether BMO removability for K-quasiregular mappings and (L
) removability for K-quasiregular mappings are indeed different problems. In this paper, we give a series of examples answering the positive Question 4.2 in [6], at the same time proving sharpness in two different senses of Theorem 1.1 in [6], and also giving examples that would yield sharpness in those two different senses as well for the absolute continuity of Hausdorff measures under push-forward by quasiconformal mappings, were it to be proved.