Math Conjectures

Corona problem for the unit ball

OPENMajorOpen problemProposed c. 1962 · Standard version

Canonical statement

Let n≥2n\ge2 and m≥1m\ge1 be integers, let Bn={z∈Cn:∥z∥2<1}\mathbb B^n=\{z\in\mathbb C^n:\|z\|_2<1\}, and let H∞(Bn)H^\infty(\mathbb B^n) be the algebra of bounded holomorphic functions on Bn\mathbb B^n. If f1,…,fm∈H∞(Bn)f_1,\ldots,f_m\in H^\infty(\mathbb B^n) satisfy
inf⁡z∈Bn∑j=1m∣fj(z)∣>0, \inf_{z\in\mathbb B^n}\sum_{j=1}^m|f_j(z)|>0 ,
then there exist g1,…,gm∈H∞(Bn)g_1,\ldots,g_m\in H^\infty(\mathbb B^n) such that ∑j=1mfj(z)gj(z)=1\sum_{j=1}^m f_j(z)g_j(z)=1 for every z∈Bnz\in\mathbb B^n.
View source LaTeX
Let \(n\ge2\) and \(m\ge1\) be integers, let \(\mathbb B^n=\{z\in\mathbb C^n:\|z\|_2<1\}\), and let \(H^\infty(\mathbb B^n)\) be the algebra of bounded holomorphic functions on \(\mathbb B^n\). If \(f_1,\ldots,f_m\in H^\infty(\mathbb B^n)\) satisfy \[ \inf_{z\in\mathbb B^n}\sum_{j=1}^m|f_j(z)|>0 , \] then there exist \(g_1,\ldots,g_m\in H^\infty(\mathbb B^n)\) such that \(\sum_{j=1}^m f_j(z)g_j(z)=1\) for every \(z\in\mathbb B^n\).
ANAL-FA-012:[Carleson1962Corona][Krantz2008CoronaSeveralVariables][AmarBrunaNicolau2017CoronaSurvey]
Analysis and functional analysis

Notes

The corona problem for the unit ball asks whether, given bounded holomorphic functions f1,…,fmf_1,\dots,f_m on Bn⊂Cn\mathbb B^n\subset\mathbb C^n with ∑j∣fj∣\sum_j|f_j| bounded below, one can always solve the Bézout equation ∑jfjgj=1\sum_j f_jg_j=1 with bounded holomorphic gjg_j; equivalently, whether Bn\mathbb B^n is dense in the maximal ideal space of H∞(Bn)H^\infty(\mathbb B^n), leaving no "corona". The several-variable question was raised soon after Carleson proved the one-dimensional case, the corona theorem for the unit disc, in 1962, so the traditional date is approximate [Carleson1962Corona].

In one variable Carleson's theorem settles the matter completely [Carleson1962Corona]. In higher dimensions, positive results are known for related multiplier algebras and for special classes of data, and much of the theory of Carleson measures has been developed with this problem in view [AmarBrunaNicolau2017CoronaSurvey]; general background on several-variable function theory can be found in Krantz [Krantz2008CoronaSeveralVariables].

For n≥2n\ge2 the H∞H^\infty corona problem remains unresolved on the ball, and likewise on the polydisk: no counterexample is known, and no current technique reaches the full H∞H^\infty statement.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.