Commutators Of Generalized Fractional Integral Operator In A Number Of Space Bounded

[b,Tl] is the commutator generated by generalized fractional integral opertor Tl and the Lipschitz function b(x). In this paper we shall consider the boundedness of these commutators on Lebesgue spaces, Hardy spaces and Herz-type Hardy spaces .In chapter one, we shall establish the boundedness on Hardy spaces for the commutator [b,Tl] with b and will prove that, this commutator is bounded from Hardy spaces to the weak Lebesgue spaces on the endpoint. The main results in the first chapter can be stated as follows:Theorem 1.1.1 Let Lipβ(Rn),and Tl be a (θ, N')-type fractional integral operator with N ≤ [n(1/p-1)]. Let .s be a noimegative integer and s ≤ N ≤ [n(1/p-1)]. If the noimegative function θ satisfiesthen [b,Tl] is bounded from Hp(Rn] into Lq(Rn).Theorem 1.1.2 Let, 0 < β < 1, 0 < l < n -β, b ∈ Lipβ(Rn), and T; be a (θ, N)-type fractional integral operator. If the noimegative function 0 satisfiesthen [b,Tl] is bounded from H n(n+β)(Rn] into WLn(n-1) (Rn).In chapter two, we shall continue to estalish the boundedness on Hera-type Hardy spaces for the commutator [b, Tl] with b ∈ Lipβ(0 <β≤1) and will prove that this commutator is bounded from Herz-type Hardy spaces to the weak Hera spaces on the endpoint. The main results hi this chapter can be stated as follows:Theorem 2.1.1 Let b Lip (Rn), , and Tl be a ( , N)-type fractional integral operator with N . If satisfiesthen [b,Tl] is bounded from H into (Rn).Theorem 2.1.2 Let b ,and Tl be a ( )-type fractional integral operator. If 0 satisfiesthen [ b, Tl] is bounded from H (Rn) into WIn diaper three, we shall consider the bonndedness on Lebesgne spaces for the commutator [b,Tl] with . The main result in this chapter is:Theorem 3.1.1 Let and T/ be a ( , N)-type fractional integral operator. If 9 satisfiesthen [b,Tl] is bounded from Lp(Rn) into (Rn).