Modern harmonic analysis: singular integral operators, function spaces and applications

In this thesis, we aim to study Besov spaces associated with operators and our work has 2 new parts: one part is to extend certain results of [BDY] to different type of heat kernels not included in [BDY] and another part is to obtain molecular and atomic decompositions for Besov spaces for a larger range of indices. Recently, in [BDY] the authors investigated the theory of Besov spaces associated to operators whose heat kernel satisfies an upper bound of Poisson type on the space of polynomial upper bound on volume growth. They also carried out that by different choices of operators L, they can recover most of the classical Besov spaces. Moreover, in some particular choices of L, they obtain new Besov spaces. In the first new part of this thesis, we aim to extend certain results in [BDY] to a more general setting when the underlying space can have different dimensions at 0 and infinity. In the second new part of this thesis, the main aim is to lay out the theory of Besov spaces associated to operators L whose heat kernel satisfies the Gaussian upper bounds on spaces of homogeneous type. The main contribution is to investigate the atomic and molecular decompositions of functions in the new Besov spaces. We also carry out the study that depending on the choice of L, our new Besov spaces may coincide with or may be properly larger than the classical Besov spaces in the space of homogeneous type. Finally, the behaviour of fractional integrals and spectral multipliers on the new Besov spaces is also investigated.