湖南大学 硕士学位论文 带可变Caldern-Zygmund核的多线性奇异积分算子的有界性研 究 姓名：徐红 申请学位级别：硕士 专业：基础数学 指导教师：刘凤喆 20070515 ? ? ?Calder on-Zygmund?T? ?. ?. ?,? ?. ?Calder on-Zygmund?BMO ?TA?Lp?Morrey?. ?Calder on-Zygmund?BMO? ?TA?Hardy?HerzHardy? ?. ?Calder on-Zygmund?Lipschitz? ?TA? TA?Hardy?Herz Hardy? ?. ?,?Calder on-Zygmund? TA? Hardy?Herz Hardy?. ? :?;?Calder on-Zygmund?; BMO?; Morrey?;Hardy?; Herz?; Herz Hardy?. ? ? Abstract In this paper, we mainly study the boundedness of the multilinear integral op- erator generated by the singular integral operator with variable Calder on-Zygmund kernel T and locally integrable functions. The content of this paper is divided into fi ve parts. In the fi rst part of this paper,we briefl y introduce the background of the pa- per and their signifi cance in theory and practice. we also introduce some signs, defi nitions and lemmas. In the second part, we prove the weighted boundedness for the multilinear singular integral operators with variable Calder on-Zygmund kernel generated by the singular integral operator with variable Calder on-Zygmund kernel T and BMO functions on Lpand Morrey spaces. In the third part,the boundedness for the multilinear singular integral oper- ators with variable Calder on-Zygmund kernel generated by the singular integral operator with variable Calder on-Zygmund kernel T and BMO functions on Hardy and Herz type Hardy spaces are obtained. In the fourth part of this paper,we prove the continuity for the multilinear oper- ators generated by the singular integral operators with variable Calder on-Zygmund kernel and Lipschitz functions on some Hardy and Herz-type spaces. Finally, we obtain the weighted endpoint estimates for the multilinear singular integral operators with variable Calder on-Zygmund kernel on Hardy and Herz type Hardy spaces. Key Words:Multilinear singular integral operator; Variable Calder on-Zygmund kernel; BMO space; Morrey space; Hardy space; Herz space; Herz Hardy space. ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 1? ? 2? (?X?) ? ? ? ? I ? ?1? 1.1? ? ?.? ?A.P.Calder on?A.Zygmund? ?,? ?(?1?11).? ?,?.? ? ?.?1?, Calder on?Zygmund? ?.?9?,? ?BMO?.?11?,? ?BMO?. Cohen? Gosselin(?3?5)?Lp(p 1)?.? ?Calder on-Zygmund? ?. 1.2? ?, Q?Rn?.?f, f? Sharp? f#(x) = sup xQ 1 |Q| Z Q |f(y) fQ|dy, ?fQ= |Q|1 R Q f(x)dx.?10,? f#(x) sup xQ inf cC 1 |Q| Z Q |f(y) c|dy. ?f?BMO(Rn)?f#?L(Rn)?|f|BMO= |f#|L.?M Hardy Littlewood?,? M(f)(x) = sup xQ |Q|1 Z Q |f(y)|dy, ?Mp(f) = (M(fp)1/p, 0 0,Lipschitz?Lip(Rn)?f?(? 12) |f|Lip=sup x,hRn, h0 |f(x + h) f(x)|/|h| 1,s 1, ?H older? 1 |Q| Z Q f1(x)f2(x)dx ? 1 |Q| Z Q (f1(x)rdx ?1/r? 1 |Q| Z Q (f1(x)sdx ?1/s . ?1.2.2(?10)?w A1,?w?H older?,? 1 1)?.?Calder on-Zygmund? ?TA?Lp?Morrey?. ?2.1.1(5)?ARn?, DA Lq(Rn)? | = m?q n.? |Rm(A;x,y)| C|x y|m X |=m ? 1 |Q(x,y)| Z Q(x,y) |DA(z)|qdz ?1/q , ? Q?x?5n|x y|?. ?2.1.1?1 0.? (S A hk(f) #( x) C l Y j=1 X |j|=mj |DjAj|BMO kn/2Mq(f)( x). ? S A hk(f2)(x) S A hk(f2)(x0) = Z Rn ? Yhk(x y) |x y|n+m Yhk(x0 y) |x0 y|n+m ? 2 Y j=1 Rmj(Aj;x,y)f2(y)dy + Z Rn ? Rm1(A1;x,y) Rm1(A1;x0,y) ? Rm2(A2;x,y) |x0 y|m+n Yhk(x0 y)f2(y)dy + Z Rn ? Rm2(A2;x,y) Rm2(A2;x0,y) ? Rm1(A1;x0,y) |x0 y|m+n Yhk(x0 y)f2(y)dy X |1|=m1 1 1! Z Rn Rm2( A2;x,y)(x y)1 |x y|m+n Yhk(x y) Rm2( A2;x0,y)(x0 y)1 |x0 y|m Yhk(x0 y)D1 A1(y)f2(y)dy X |2|=m2 1 2! Z Rn Rm1( A1;x,y)(x y)2 |x y|m+n Yhk(x y) Rm1( A1;x0,y)(x0 y)2 |x0 y|m+n Yhk(x0 y)D2 A2(y)f2(y)dy + X |1|=m1, |2|=m2 1 1!2! Z Rn (x y) 1+2 |x y|m+n Yhk(x y) (x0 y)1+2 |x0 y|m+n Yhk(x0 y)D1 A1(y)D2 A2(y)f2(y)dy =I(1) 5 + I(2) 5 + I(3) 5 + I(4) 5 + I(5) 5 + I(6) 5 . ?2.1.1?(?10) |bQ1 bQ2| C log(|Q2|/|Q1|)|b|BMO,Q1 Q2, ?x Q?y 2i+1Q 2iQ,? |Rm(A;x,y)|C|x y|m X |=m (|DA|BMO+ |(DA) Q(x,y) (D A) Q|) 9 ?Calder on-Zygmund?