The Boundedness Of Some Singular Integral Operators

This PH. D thesis focuses on the boundedness of some important singular integral operators on BMO space, Campanato space, BLO space, and Hωρspace and so on. The Lρboundedness of these operators has been studied thoroughly.This paper consists of senven chapters. In Chapter 1, we present the defini-tions of BMO space, Campanato space, BLO space, and Hωρspace and so on. We also give the basic properties of these spaces and the main works of this paper.In Chapter 2, we consider the behaviors of a class of parametric Marcinkiewicz integralsμΩρ,μΩ,λ*,ρandμΩ,Sρon BMO(Rn) and Campanato spaces with complex parameter p and the kernelΩin L log+ L(Sn-1). Under certain weak regularity condition on Q, we proved that if f belongs to BMO(Rn) or to a certain Cam-panato space, then [μΩ,λ*,ρ(f)]2, [μΩ,Sρ(f)]2 and [μΩρ(f)]2 are either infinite everywhere or finite almost everywhere, and in the latter case, some kind of boundedness are also established.In Chapter 3, we studied the boundedness of maximal singular integral oper-ator, and generalized Hu and Zhang's results to different.In Chapter 4, we consider the fractional parametric Marcinkiewicz integralμΩ,αρwith variable kernel. Without any smoothness assumption on the kernelΩ, we show thatμΩ,αρ is bounded from L2n/n+2α(Rn) to L2(Rn).In Chapter 5, we consider the boundedness ofμΩ,b,αρon Hardy type space Hp/b(Rn). Where 00,Σi=1 mβi=β,0≤α<β< 1.(see definition 1.2.5.) In Chapter 6,we will prove the Hwp-Lwp boundedness of paramet ric Marcinkiewicz integral whereω∈Ap.For f∈Lr(Rn)∩BMO,the following inequality‖f‖p≤Cn,p‖f‖rr/p‖f‖BMO,1-r/p,1≤r≤p<∞was proved in[1].In Chapter 7,we will give another two inequalities which contains Chen and Zhu[56]'s results. As a consequence,the following Kozono-Tauiuchi's inequality([57])‖fg‖r≤Cn,r(‖f‖r‖g‖BMO+‖g‖r‖f‖BMO),0