Sharp Examples for Planar Quasiconformal Distortion of Hausdorff Measures and Removability

Abstract

In the celebrated paper [Ast94], Astala showed optimal area distortion bounds and

Sharp nonremovability examples for Hölder continuous quasiregular mappings in the plane

“…It is instructive to recall the following elementary Lemma in [19], which we prove for the reader's convenience. Hence, by the above Lemma, in order to fill a very big proportion of the area of the unit disk D with smaller disks we are forced to consider disks of different radii.…”

“…We remark two points in this argument. First, it is precisely the distortion property (1.4) for H d and H d ′ obtained in [19] what allows us to get non removable sets at the critical dimension d (and even with finite H d measure.) Second, several technical difficulties will arise when computing the Hölder exponent of φ, because of the fact that the set E is highly nonregular.…”

“…Recently, it was shown by Uriarte-Tuero [19] that equality at (1.2) may be attained even at the level of measures. More precisely, Question 4.2 in [3] asked whether there exists, for every K ≥ 1, a compact set E with 0 < H 2 K+1 (E) < ∞, such that E is not removable for some K-quasiregular functions in BM O(C).…”

“…More precisely, Question 4.2 in [3] asked whether there exists, for every K ≥ 1, a compact set E with 0 < H 2 K+1 (E) < ∞, such that E is not removable for some K-quasiregular functions in BM O(C). In [19], the author gives an affirmative answer to this question by building a highly non-selfsimilar and non-uniformly distributed Cantor-type set E and a K-quasiconformal mapping φ such that…”

“…In section 2, we recall from [19] how to construct, for any 0 < t < 2 and K > 1, a K-quasiconformal mapping φ and a set E ⊂ C such that 0 < H t (E) < ∞ and 0 < H t ′ (φ(E)) < ∞, t ′ = 2Kt 2+(K−1)t . In section 3, we prove that this K-quasiconformal mapping φ is locally Hölder continuous with exponent t t ′ .…”