Popular presentations of Carleson's results

The Corona theorem and Carleson measure

In mathematical literature, the word theorem is synonymous with the perhaps more easily understandable main result. These main results tend to be given names, Fermat's Last Theorem, Abel's Addition Theorem, the Fundamental Theorem of Algebra or more everyday names like the Triangle Inequality, the Spectral Theorem or the Corona theorem.

Carleson's Corona theorem refers to the solar corona, the ring of glowing matter around the sun that can only be observed during total solar eclipses. Carleson has never been an expert on the sun's interior (or exterior) life, but he has proved a difficult theorem that has been given its name from the sun. It was the Japanese mathematician Kakutani who came up with a conjecture that became known as the Corona problem in the beginning of the 1940s. A conjecture is not the same as a theorem; a conjecture has not been proven! This is a trick that mathematicians occasionally resort to. In some situations, they are convinced that a result is correct, even though they cannot manage to proof it. All examples confirm it, and in many special cases the result can actually be formally proven, but they are unable to come up with the final reasoning that is necessary in order to pronounce the result proven. What do they do then? Put everything away in a drawer and try to forget it? No, they pose a conjecture that they publish in the same way as they would fully proven results, and then they call it a conjecture. If he/she is lucky, posterity will name the conjecture after the conjecturer, even if someone completely different actually manages to prove the result.

This then was the fate of the Corona problem, a conjecture everyone believed, but no one could prove. What was it about? The Corona problem considers certain functions defined in a circular disk. The edge of this disk is a circle. If these functions behave properly within the circle, how many curls can they then “come up with” in the actual circle? Carleson's theorem gives an answer to this question. And the analogy to the corona? The circular disk is the sun, and what occurs at the edge, i.e. of the circle, corresponds to the corona.

This result of Carleson is also an example of how the solution of a problem has had an effect on other problems. In Carleson's proof of the Corona problem, he introduces a measure. In this context, a measure is a way of assigning a positive number to a given set. For example, we can define a measure for a series of numbers by assigning an interval its length. Or that a set in a plane is assigned its area. Carleson needed to measure the length of certain curves he constructed on the circular disk and introduced a measure for this purpose. In posterity, this measure has obviously been called the Carleson measure, and it has proven to be an unusually useful aid in many fields of mathematics.

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