Multipliers and Bourgain algebras of \(H^ \infty+C\) on the polydisk (Q1918393)

Let \(U^2\) be the unit bidisk and \(T^2\) the corresponding torus. Let \(H^\infty (U^2)\) denote the space of bounded holomorphic functions on \(U^2\), the corresponding space of radial limit functions being denoted \(H^\infty (T^2)\). The space of multipliers \({\mathcal M}\) is defined to be \(\{f\in L^\infty (T^2): f \cdot (H^\infty (T^2)+ C(T^2)) \subset H^\infty (T^2)+ C(T^2) \}\). The authors show that \({\mathcal M}\) is a proper subalgebra between \(A(T^2)\) and \(H^\infty (T^2)\). Moreover, a function \(f\in H^\infty (T^2)\) is in \({\mathcal M}\) if and only if its Poisson kernel \(\widetilde f\) has a continuous extension to \(\overline U^2 \backslash T^2\). Also, for a closed subspace \(Y\) of a commutative Banach algebra \(X\), the Bourgain algebra \(Y_b\) is defined to be the set \(\{f\in X: |ff_n+Y |\to 0\) for every weakly null sequence \((f_n)\) in \(Y\}\) [\textit{J. Cima} and \textit{R. M. Timoney}, Mich. Math. J. 34, 99-104 (1987; Zbl 0617.46058)]. Using their characterization of \({\mathcal M}\), the authors calculate the Bourgain algebra \(Y_b\) and higher Bourgain algebras (e.g. \((Y_b)_b= Y_{bb})\), for spaces of the form \(H^\infty +C\) relative to \(T^2\) and \(U^2\).