In this paper we will give an improved estimate in the vector valued corona theorem for the unit disk. Our proof will be split into two parts: first, we will resurrect the operator corona theorem from the 1970s, and then we will use a Hilbert space argument, based on a version of Wolff's ideas. Since we use Hilbert space methods, the Riesz representation theorem replaces the usual " ยฏ-techniques.
In 1962 Carleson determined when a finitely generated ideal in H . (D) is actually all of H . (D), by giving a function theory condition on the generators. Namely, I(f 1 , ..., f m )=H . (D) if and only if there exists an e > 0, so that ; m i=1 |f i (z)| 2 \ e 2 for all z ยฅ D. Actually, more was shown, as a bound for the size of solutions was given. More precisely, if
e 2 for all z ยฅ D, then there exists a B(e, m) < . and there exist {g i } m i=1 โฆ H . (D) with ; m i=1 f i g i =1 and ; m 1 |g i (z)| 2 [ B(e, m) for all z ยฅ D. This paper [5] spawned numerous investigations in many directions, but we will mention only those most closely related to our work. For the scalar case above, Hormander introduced " ยฏ-methods, culminating in Wolff's surprising proof of the corona theorem. (See [8].) Rosenblum [14] and, independently, Tolokonnikov [17] removed the dependency on m. This was fine-tuned by Uchiyama [20] to yield the estimate (for e small) B(e) [ C e 2 ln 1 e 2 , Ca universal constant.