In this Master's thesis we study global and local Tb theorems for square functions with L^2 testing conditions. Our setting is very general and involves the very latest research methods in the field. We work in metric spaces equipped with a non-homogeneous measure and the square function operator domain being the L^2-space.
The square functions are a certain type of singular integral operators involving so called Littlewood-Paley integral kernels of form
...,
where
...
Here for every t > 0 the kernel ... , is assumed to satisfy the so called size- and Hölder-estimates. These estimates characterise the growth properties of the kernel.
The main theorems we prove, global version of T1 and local version of Tb, characterise the L^2 boundedness of the square functions on certain testing conditions. In the T1 theorem we test the boundedness of the operator with constant function 1 over all cubes. The Tb testing conditions involve a family of testing functions. The proofs of these theorems require dyadic cubes and -grids which are constructed on chapter 2 and the involved dyadic methods, which are presented within the proofs.
The main tools in this Master's thesis are: the basic L^p-space methods e.g. maximal functions, dyadic grids in metric spaces, randomisation of metric dyadic cubes, standard and adapted martingale transforms, Carleson estimates, dyadic summation arguments and probabilistic arguments related to dyadic cubes (including the bad/good cubes).
