This is an announcement for the paper "Extension of functions with small oscillation" by Denny H. Leung and Wee-Kee Tang.
Abstract: A classical theorem of Kuratowski says that every Baire one function on a G_\delta subspace of a Polish (= separable completely metrizable) space X can be extended to a Baire one function on X. Kechris and Louveau introduced a finer gradation of Baire one functions into small Baire classes. A Baire one function f is assigned into a class in this heirarchy depending on its oscillation index \beta(f). We prove a refinement of Kuratowski's theorem: if Y is a subspace of a metric space X and f is a real-valued function on Y such that \beta_{Y}(f)<\omega^{\alpha}, \alpha < \omega_1, then f has an extension F onto X so that \beta_X(F)is not more than \omega^{\alpha}. We also show that if f is a continuous real valued function on Y, then f has an extension F onto X so that \beta_{X}(F)is not more than 3. An example is constructed to show that this result is optimal.
Archive classification: Classical Analysis and ODEs; Functional Analysis
Mathematics Subject Classification: 26A21; 03E15, 54C30
The source file(s), DLeungWTangBaire1Ext.tex: 47118 bytes, is(are) stored in gzipped form as 0505168.gz with size 13kb. The corresponding postcript file has gzipped size 71kb.
Submitted from: wktang@nie.edu.sg
The paper may be downloaded from the archive by web browser from URL
http://front.math.ucdavis.edu/math.CA/0505168
or
http://arXiv.org/abs/math.CA/0505168
or by email in unzipped form by transmitting an empty message with subject line
uget 0505168
or in gzipped form by using subject line
get 0505168
to: math@arXiv.org.