Abstract:
A classical problem in complex analysis is to characterize the removable sets
for various classes of analytic functions: H\"{o}lder, Lipschitz, BMO, bounded
(this last case gives rise to the analytic capacity and the Painlev\'{e}
problem which has been recently solved by Tolsa.)
One can ask the same questions in the setting of K-quasiregular maps (since
they are a K-quasiconformal map followed by an analytic map.) The
K-quasiregular maps are precisely the solutions to the Beltrami equation,
and have found applications in areas such as nonlinear elasticity,
and electrical impedance tomography.
Most of the bounded case was dealt with in a joint paper with K. Astala,
A. Clop, J.Mateu and J.Orobitg, [ACMOUT]. The BMO case was dealt with in
[ACMOUT] except for a gap at the critical dimension (Question 4.2 in
[ACMOUT].) I answered the question filling the gap in [UT]. The Lipschitz
case was dealt with by A. Clop, as well as most of the H\"{o}lder case,
where again a gap at the critical dimension was left. In a joint paper
with A. Clop [CUT] we closed the gap.
I will summarize the results and give some ideas of the proofs in the above
papers. The talk will be self-contained.
References:
[ACMOUT] Kari Astala, Albert Clop, Joan Mateu, Joan Orobitg and Ignacio
Uriarte-Tuero. Distortion of Hausdorff measures and improved Painlev\'{e}
removability for bounded quasiregular mappings. Duke Math J., to appear.
[CUT] Albert Clop and Ignacio Uriarte-Tuero. Sharp Nonremovability Examples for
H\"{o}lder continuous quasiregular mappings in the plane. Submitted.
[UT] Ignacio Uriarte-Tuero. Sharp Examples for Planar Quasiconformal Distortion
of Hausdorff Measures and Removability. Submitted. |