Some remarks to the corona theorem (Q2849058)

Let \(A\) and \(B\) be Banach spaces. The space \(H^p(B)\) will denote the Hardy space of analytic functions on the unit disc taking values in \(B\), and \(H^\infty(\mathcal{L}(A,B))\) denotes the bounded analytic functions on the unit disc taking values in \(\mathcal{L}(A,B)\), the linear operators from \(A\) to \(B\). For \(1\leq p\leq\infty\) the \(L^p\) corona problem with data \(F\in H^\infty(\mathcal{L}(A,B))\) is solvable with constant \(C\) if for every function \(x\in H^p(B)\) there exists a function \(y\in H^p(A)\) such that NEWLINE\[NEWLINE F(z)y(z)=x(z)\quad \forall z\in\mathbb{D} NEWLINE\]NEWLINE and \(\left\| y\right\|_{H^p(A)}\leq C\left\| x\right\|_{H^p(B)}\).NEWLINENEWLINEThe authors of this paper are interested in the \(L^p\) corona problem on the disc and their results can be interpreted as saying if the \(L^p\) corona problem with data \(F\) is solvable with constant \(C\) for one value of \(p\), \(1\leq p\leq\infty\), then it is solvable for all values of \(p\).