Corona theorem

Remarks

The hypothesis is necessary: if $\sum f_j g_j = 1$ with $\abs{g_j} \le C$, then $\sum \abs{f_j} \ge 1/C$ on $\mathbb{D}$. The theorem is equivalent to the density of $\mathbb{D}$ in the space of maximal ideals of the Banach algebra $H^\infty$. Here $z \in \mathbb{D}$ corresponds to the ideal of functions vanishing at $z$. The name comes from this picture: without the theorem, there could be a "corona" of maximal ideals away from the closure of the disc.

Proof idea: Carleson proved the theorem in 1962 with a construction that led to Carleson measures. A shorter proof due to Wolff solves the problem first with smooth functions, then corrects them by solving a $\bar\partial$-equation with bounded solutions, using a Carleson measure estimate.

Example: for $n = 1$ the theorem says that if $\abs{f_1} \ge \delta$ on $\mathbb{D}$, then $1/f_1$ is bounded and holomorphic, which is clear.

Going deeper

For a student

In Banach algebra language, the theorem identifies where the disc sits inside the maximal ideal space of $H^\infty$: it is dense. Its proof introduced Carleson measures and interpolating sequences, which became central tools in function theory. The analogue for the unit ball in $\C^n$ with $n \ge 2$ is a well known open problem.