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 Banachtopological algebra F on the unit circle T, we estimate solutions g_jA to the corresponding Bezout equation _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.