Resumen de Bounding smooth solutions of Bezout equations

Nikolai Nikolski

• Given data f=(f1,f2,…,fn) in the holomorphic part A=F+ of a symmetric Banach\slash topological algebra F on the unit circle T, we estimate solutions gj∈A to the corresponding Bezout equation ∑nj=1gjfj=1 in terms of the lower spectral parameter δ, 0<δ≤|f(z)|, and an inversion controlling function c1(δ,F) for the algebra F. A scheme developed issues from an analysis of the famous Uchiyama-Wolff proof to the Carleson corona theorem and includes examples of algebras of “smooth” functions, as Beurling-Sobolev, Lipschitz, or Wiener-Dirichlet algebras. There is no real “corona problem” in this setting, the issue is in the growth rate of the upper bound for ∥g∥An as δ→0 and in numerical values of the quantities that occur, which are determined as accurately as possible.