Estimates in corona theorems for some subalgebras of H∞

Abstract

If n is a non-negative integer, then denote by ∂^-nH^∞ the space of all complex-valued functions f defined on $\mathbb{D}$ such that f, f⁽¹⁾, f⁽²⁾,..., f⁽ⁿ⁾ belong to H^∞, with the norm$\|f\|=\sum_{j=0}^{n}\frac{1}{j!}\|f^{(j)}\|_{\infty}.$We prove bounds on the solution in the corona problem for ∂^-nH^∞. As corollaries, we obtain estimates in the corona theorem also for some other subalgebras of the Hardy space H^∞.