The $T(1)$ theorem of David and Journé is one of the most remarkable theorems in harmonic analysis. The theorem reduces the study of $L^{p}$ boundedness of a singular integral operator, $T$ to testing a 'testing condition', that is, verifying $T(1)$ is in the space $B M O$. A simplistic view of these theorems is that they shift the task of verifying boundedness for all functions (globally) to that of verifying a condition on all cubes. More general testing conditions, e.g. 'local $T(b)$' conditions, allow one to adapt the testing function to the cube and/or weaken conditions on the operator. These 'local $T(b)$ theorems' are an important ingredient to the initial solution to the Kato problem.
The project will introduce the concepts of $T(1) / T(b)$ theory for singular integrals, Littlewood-Paley theory, Carleson measures and stopping time arguments. The goal is to present the 'original' proof of the Kato problem and, possibly, look at more recent developments.[-]