| In the present dissertation, we address several questions concerning boundedness properties of singular and maximal singular integral operators with rough homogeneous kernels.; In the first chapter, we give a necessary and sufficient condition for an integrable compactly supported function with mean value zero on the line to belong to the Hardy space H1(R 1) and we use this result to obtain a new characterization of H1(S1). We also give some applications in the context of singular integrals.; In the second chapter, we obtain Lp bounds for singular and maximal singular integrals with homogeneous kernels satisfying various size conditions.; In the third chapter, we address the problem of weak type (1, 1) boundedness of Calderón-Zygmund operators with homogeneous rough kernels. We prove uniform weak type (1, 1) bounds for operators whose kernels which can be written as sums of “equidistributed” atoms on the circle S 1. |
