Some remarks to the corona theorem

Abstract:

With the help of a fixed point theorem, in §1 it is shown that the so-called $L^{\infty }$- and $L^p$-corona problems are equivalent in the general situation. This equivalence extends to the case where $L^p$ is replaced by a more or less arbitrary Banach lattice of measurable functions on the circle. In §2, the corona theorem for $\ell ^2$-valued analytic functions is exploited to give a new proof for the existence of an analytic partition of unity subordinate to a weight with logarithm in BMO. In §3, simple observations are presented that make it possible to pass from one sequence space to another in $L^{\infty }$-estimates for solutions of corona problems.