Undergraduate Honors Thesis Presentation: The Tb Representation Theorem

Abstract: We want to study harmonic analysis, specifically the methods of dyadic harmonic analysis. These are discrete methods that nevertheless apply to continuous problems via various different means. We will study the so-called probabilistic-dyadic methods, where randomized dyadic grids are used, and how these methods are used to analyze singular integrals. The aim is to present dyadic cubes, the way to randomize dyadic cubes and, as the main result, characterize the boundedness of singular integrals using a somewhat refined version of existing representation theorems. 

The idea of Tb theorems is that the singular integral is tested on just a single, suitably non-degenerate, function b, and this is enough for boundedness on Lebesgue spaces.  On the other hand, representation theorems are even finer results that faithfully represent a singular integral as an average of dyadic operators. A representation theorem usually directly implies a T1 theorem (a Tb theorem with b=1). Historically, Tb theorems with a more general b have been proved in a slightly different way, not via an explicit representation theorem. The idea here is to prove a Tb theorem, not just a T1 theorem, but do it using a representation theorem. This requires adapting some of the latest proofs of representation theorems to explicitly yield b-adapted model operators.

Host: Henri Martikainen