## Duke Mathematical Journal

### Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings

#### Abstract

The classical Painlevé theorem tells us that sets of zero length are removable for bounded analytic functions, while (some) sets of positive length are not. For general $K$-quasiregular mappings in planar domains, the corresponding critical dimension is ${2}/({K+1})$. We show that when $K>1$, unexpectedly one has improved removability. More precisely, we prove that sets $E$ of $\sigma$-finite Hausdorff $({2}/({K+1}))$-measure are removable for bounded $K$-quasiregular mappings. On the other hand, $\dim(E) = {2}/({K+1})$ is not enough to guarantee this property.

We also study absolute continuity properties of pullbacks of Hausdorff measures under $K$-quasiconformal mappings: in particular, at the relevant dimensions $1$ and ${2}/({K+1})$. For general Hausdorff measures ${\cal H}^t$, $0 \lt t \lt 2$, we reduce the absolute continuity properties to an open question on conformal mappings (see Conjecture 2.3)

#### Article information

Source
Duke Math. J., Volume 141, Number 3 (2008), 539-571.

Dates
First available in Project Euclid: 15 February 2008

https://projecteuclid.org/euclid.dmj/1203087637

Digital Object Identifier
doi:10.1215/00127094-2007-005

Mathematical Reviews number (MathSciNet)
MR2387431

Zentralblatt MATH identifier
1140.30009

#### Citation

Astala, K.; Clop, A.; Mateu, J.; Orobitg, J.; Uriarte-Tuero, I. Distortion of Hausdorff measures and improved Painlevé removability for quasiregular mappings. Duke Math. J. 141 (2008), no. 3, 539--571. doi:10.1215/00127094-2007-005. https://projecteuclid.org/euclid.dmj/1203087637

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