The Corona Problem, Carleson Measures, and Applications - Saracco, Alberto (Mathematical Analysis and Applications, 2018)
Carleson measures were introduced by Lennart Carleson in 1962 to solve an interpolation problem about bounded holomorphic function called the corona problem. Since then Carleson measures have been both a powerful tool and a mathematical object worth of study per se and research on the corona problem or Carleson measures has gone a long way in the last 50 years. In this chapter, we will review the developments and generalizations of the corona problem, the results on Carleson measures themselves and some applications of Carleson measures, in several different settings, starting from the disc in C (where the corona problem was originally set) arriving to the unit ball in C n , to bounded strongly pseudoconvex domains and even to domains in the quaternionic space. Both the corona problem and the Carleson measures still need investigation, as many open problems have not been solved yet. The open problems are highlighted with a bold font, for the reader's convenience.