The Corona Problem

These are the lecture notes generated for the CIMPA Summer School: Real and Complex Analysis with Applications to other Sciences that took place in Buea, Cameroon May 2 13, 2011. The topic of the series of lectures that were given focused on the Corona Problem in complex analysis. This problem can serve as a point of entry to numerous areas of analysis: complex analysis, harmonic analysis, operator theory, and real analysis. In fact, mastering many of the ideas that appear in the proof of the Corona Theorem will be immensely beneficial for students of analysis. There were three lectures given during the CIMPA school. Additionally, there were two problem sessions associated with the lectures so that the students could gain some mastery of the material being presented. The course notes first introduce most of the topics necessary for the Hardy space and then turned to the necessary background for the Corona problem. The topics covered in the course include the following • Definitions of these Spaces; • Computations of their Reproducing Kernels; • Definitions of their Carleson measures and Geometric Characterizations; • Corona Theorems. Based on my personal interests, we focused much more on the function theory and harmonic analysis aspects associated with these spaces.