Littlewood-Paley Operators With Rough Kernels On The Weighted Spaces

The core of this dissertation, which consists of three chapters, will mainly deal with the boundedness of Littlewood-Paley operators with rough kernels on some weighted spaces.In the chapter 1, with the help of the weighted inequalities and the boundedness on the weighted Lebesgue spaces, the weak type estimates of the square function with rough kernel and Littlewood-Paley gλ*-function are obtained respectively on the weighted Morrey spaces Lp,λ(ω) as the kernel Ω2 ∈Lq(Sn-1)(l<q≤∞) is homoge-neous of degree zero and has mean value zero on Sn-1.In the chapter 2, as the kernel Ω∈Ls(5n-1)(l< s≤∞) is homogeneous of degree zero and has mean value zero on Sn-1, we prove that Litttlewood-Paley area integrals with rough kernel on the weighted Herz spaces are bounded.In the chapter 3, as the kernel Ω∈ Ls(5n-1)(l< s≤∞) is homogeneous of degree zero and has mean value zero on Sn-1, the weak type estimates of the Litttlewood-Paley area integrals with rough kernel on the weighted Herz spaces are obtained.