Singular Integral And Fractional Commutators Boundedness

In this paper, the boundedness of multilinear commutators of Calderon-Zygmund operator and fractional integral are studied.In chapter 1. the author studies the boundedness of multilinear commutators of Calderon-Zygrnund operator on Hardy-type spaces . Let b=(b1, b2, bm), bi BMO. 1 < i < m. the multilinear commutators are defined bywhere K(x,y) is Calderon-Zygmund kernel. In this chapter, we investigate the boundedness of operator Tb on Hardy spaces Hb, weak Hardy spaces Hb(Rn) and Herz type Hardy spaces HkbIn chapter 2 , we study the boundedness of maximal multilinear commutators of Tb we first show that vector-valued multilinear commutatoris bounded from LPE( Rn) to LPF(Rn),1 < p <, where E, F are Banach spaces and bi BMO(E,E)(Rn), i=1, 2, m. Then, by using the previous result it is also proved that maximal multilinear commutatoris bounded on Lp(Rn) for 1 < p < and bi BMO(Rn), i= 1,2,m. In the case of p = 1, a weak-type estimate of Tb is given.In chapter 3 , the author studies the boundedness of multilinear commutators of fractional integral operator . Letbe the standard fractional integral operator. Multilinear commutators [b, Ia] with vector b = (b1,b2,bm) defined byare considered , where bi BMO(Rn),i=1,2,m. In this chapter, we give an estimate of sharp maximal operator M([b,Ia]). By using this estimation, we show that [b, Ia] is bounded from Lp(Rn) to Lq(Rn), where 1) and Herz spaces. By the later result , the author proves the boundedness of a class of multilinear commutators of Tl, on Herz spaces.