From: Brenda Sue Cook (brenda@math.missouri.edu)
Date: Mon Oct 22 2007 - 11:19:40 CDT
Harmonic Analysis Seminar
Monday, October 22, 4:40-5:30 PM
Room: GCB 209
Speaker: Ignacio Uriarte-Tuero (UMC)
Title: Sharp nonremovability examples for BMO and H\"{o}lder
quasiregular mappings
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]. 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.
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