Singular integrals with rough kernals and BMO spaces associated to operators

The purpose of this thesis is to study the new criterion for boundedness of some singular integrals. The main results of this thesis are presented in four parts. --1. Recall the Hörmander condition for boundedness of singular integrals which has been an important result of the Calderón-Zygmund theory. -- 2. Discuss a new criterion for singular integral operators to be bounded on Lp(X), 1< p < ∞ , where X is a space of homogeneous type. This criterion is an improvement of the Hörmander condition and it has had many applications in recent research of singular integrals in the last 20 years. -- 3. Discuss new function spaces which suit these operators such as BMOA spaces associated to operators. -- 4. Use these results to study the functional calculus of operators satisfying certain kernel estimates.