Bourgain algebras of ideals in $H^\infty$ generated by inner functions

Abstract

In this paper we prove that Bourgain algebra of $qX$ relative to $L^{\infty } $ contains $C$ if $X$ is backward shift invariant and $q$ is an inner function, i.e. $\left(qX,L^{\infty } \right)_{b} \supset C$.  We also studied some Bourgain algebras of  finitely generated ideals in $A$ and $H^{\infty } $.