Estimates in corona theorems for some subalgebras of $H^\infty$
Abstract
We obtain estimates in the corona theorem for the algebra of analytic functions in the unit disc whose nth derivative is bounded, and its subalgebras defined by the boundary continuity of the nth derivative. The corona theorem for such algebras is trivial and known, but corona theorem with estimates is new and much harder to prove. As an auxiliary result of independent interest we get the continuity of the best estimate in such algebras (including the classical case of the algebra of bounded analytic functions). We also show that for a fixed the best estimate is the same for all algebras we consider.
Keywords
Cite
@article{arxiv.math/0702754,
title = {Estimates in corona theorems for some subalgebras of $H^\infty$},
author = {Amol Sasane and Sergei Treil},
journal= {arXiv preprint arXiv:math/0702754},
year = {2010}
}
Comments
24 pages, 1 postscript figure. A preliminary version of this paper is available as LSE report LSE-CDAM-2006-09, see http://www.cdam.lse.ac.uk/Reports/Files/cdam-2006-09.pdf The difference with the preliminary version: minor errors/typos pointed by referee are corrected; connection with earlier work of V. Tolokonnikov is clarified; section 2.5 explaining known facts about Carleson measures is added