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Hallo,

I have read about the Corona Theorem (see link:http://en.wikipedia.org/wiki/Corona_theorem). From this one ca deduce that: Let $f_{1}, ..., f_{n}$ be holomorphic bounded functions on the unit disk, which do not all simultaniously vanish. Then there exists bounded holomorphic functions $g_{1}, ..., g_{n}$ such that $\sum_{i=1}^{n}f_{i}g_{i} = 1$. My question is: is this true in several variables? Let say, if the disk is a polydisk or some open, convex ... domain?

hapchiu

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As far as I know, when $n\geq 2$, the corona problem remains unsolved for all open convex domains in ${\mathbb C}^n$. That is, no such domain is known for which the corona problem has been decided. Certainly I remember seeing in talks by various people that the ball and the polydisc remain undecided. However, SCV is well outside my usual realm of competence so I may have misunderstood, or be unaware of the latest results. –  Yemon Choi Nov 26 '12 at 7:52
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That's correct. Unfortunately, all we currently have in, say, the ball is a BMO solution for $H^\infty$ data (see math.mcmaster.ca/~secostea/papers/BMOcorona.pdf) and a few ideas that nobody can really make work. I wouldn't be surprised if that is for a good reason and the Carleson theorem actually fails for $n>1$, but nobody knows how to construct a counterexample either. –  fedja Nov 26 '12 at 13:30
    
what do you mean by "BMO"? –  hapchiu Nov 27 '12 at 11:26
    
why does this paper not solve the posted problem? –  hapchiu Nov 27 '12 at 11:33
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Bounded mean oscillation. Those are almost bounded functions but not quite. –  fedja Nov 27 '12 at 12:47

2 Answers 2

No corona theorem fails for several variables.look here for a counterexample

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Good point. According to what I've found in some other surveys, Sibony's counterexample is a pseudoconvex domain in ${\mathbb C}^2$. However, my understanding is that the Corona Theorem remains open for the ball and the polydisc in dimensions $\geq 2$, which is what the OP seemed to be asking about. –  Yemon Choi Mar 31 '13 at 2:22

Boosting this from the comments to an answer for added visibility, since the other answer by Koushik gives only a partial picture:

As far as I know, when $n\geq 2$, the corona problem remains unsolved for all open convex domains in ${\mathbb C}^n$. That is, no such domain is known for which the corona problem has been decided. Certainly I remember seeing in talks by various people that the ball and the polydisc remain undecided.

In comments, fedja points out that a recent paper of Costea, Sawyer and Wick manages to get BMO solutions to $H^\infty$ data for the corona problem on the ball.

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