Bounding smooth solutions of Bezout equations - Nikolski, Nikolai (Mathematica Scandinavica, 2017)
Given data $f=(f_1,f_2,… ,f_n)$ in the holomorphic part $ A= F_+$ of a symmetric Banach\slash topological algebra $ F$ on the unit circle $\mathbb{T}$, we estimate solutions $ g_j\in A$ to the corresponding Bezout equation $\sum _{j=1}^ng_jf_j=1$ in terms of the lower spectral parameter δ, $0< δ ≤ |f(z)|$, and an inversion controlling function $c_1(δ ,F)$ 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 $\|g\|_{A^n}$ as $δ → 0$ and in numerical values of the quantities that occur, which are determined as accurately as possible.