Seminar

Department of Mathematics

February 25, 2008 at 3:10PM

BA6183, 40 St. George Street

DYNAMICAL SYSTEMS SEMINAR

Sharp nonremovability examples for BMO and H\"{o}lder quasiregular mappings

Ignacio Uriarte-Tuero

University of Missouri

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.) Most of the bounded case was dealt with in a joint paper with K. Astala, A. Clop, J.Mateu and J.Orobitg, [ACMOUT].

BMO is a nice substitute for $L^{\infty}$ in many analysis problems. 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. Preprint.

[UT] Ignacio Uriarte-Tuero. Sharp Examples for Planar Quasiconformal Distortion of Hausdorff Measures and Removability. Submitted.