Universal divisors in Hardy spaces

We study a division problem in the Hardy classes ${\displaystyle H^p\left(\right)}$ of the unit ball of ${\displaystyle {ℂ}^2}$ which generalizes the ${\displaystyle H^p}$ corona problem, the generators being allowed to have common zeros. MPrecisely, if S is a subset of , we study conditions on a ${\displaystyle {ℂ}^k}$-valued bounded Mholomorphic function B, with ${\displaystyle B_{\mid S}=0}$, in order that for 1 ≤ p < ∞ and any function ${\displaystyle f\in H^p\left(\right)}$ with ${\displaystyle f_{\mid S}=0}$ there is a ${\displaystyle {ℂ}^k}$-valued ${\displaystyle H^p\left(\right)}$ holomorphic function F with f = B·F, i.e. the module generated by the components of B in the Hardy class ${\displaystyle H^p\left(\right)}$ is the entire module ${\displaystyle M_S:=\{f\in H^p\left(\right):f_{\mid S}=0\}}$. As a special case, for S = ∅, we get the ${\displaystyle H^p}$ corona theorem.