Singular integrals and function spaces

The main aim of this thesis is to study the boundedness of some singular integrals on various function spaces. The main results of this thesis are presented in three parts. -- In the first part, two criteria on the Lp-weighted norm inequalities of singular integral operators with non-smooth kernels and the endpoint estimates of the commutators of these operators with BMO functions are obtained. As applications, we first studied the weighted norm inequalities of Riesz transforms associated to Schrödinger operators, Green functions and spectral multipliers and then endpoint estimates of commutators of these singular integrals with BMO functions such as the Riesz transforms, the square functions and the spectral multipliers. -- The second part is dedicated to study the Hardy spaces associated to the discrete Laplacians on graphs and applications. Some characterizations of Hardy spaces associated to operators such as the atomic characterization and the square function characterization are obtained. Then we consider the boundedness of singular integrals on these Hardy spaces. -- In the third part, we develop the theory of Hardy spaces, RBMO spaces and Calder on-Zygmund operators in the setting of nonhomogeneous spaces. Some important results are addressed in this part such as the Interpolation Theorem between Hardy spaces and RBMO spaces, the boundedness of Calder on-Zygmund operators on Hardy spaces and RBMO spaces and the Calderón-Zygmund decomposition.