Some Problems On Fractional Integral Operator,Marcinkiewicz Commutator And Multilinear Operators

It is wekk known that the fractional integrals,the commutators for the Macinkiewicz integrals and the multilinear operators play a profound and extensive role in harmonic analysis and the partial equations.In this thesis ,we'll discuss the mapping properties of these kinds of operators.our thesis consists of five chapters.Chapterl is the introductions.In chapter two ,we will dicuss the the boundedness of the fractional integral on centain weak Hardy spaces . Our main results are listed as follows:Theorem 2.4 Let 0 < α < 1, n/n+α, 1/q =1/p -α/n, and Ω ∈ Lr(Sn-1) with r > n/n-αbe homogeneous of degree zero on Rn . If the integral modulus of the continueity wr(δ) of order r of Ω satisfiesthen there is a C > 0 such thatIn chapter three ,we will discuss the boundedness of the commutators for the Marcikiewicz integrals in the Triebel-Lizorkin spaces, the main results are listded as follows:Theorem 3.2 Let 1 < p < ∞, 0<β< min(1/2,α) and b(x) ∈ ∧β, then for any f∈Lp(Rn) we haveIn charpter 4,We consider the multilinear singular integral operator defined by :Where fisatisfies certin homogenity,smoothness and symmetry ,m = m1 + m2If Aj has derivatives of order (mj - 1) belongs to BMO(Rn').The main results of this charpter are the following results ,Theorem 4.1 Suppose that f belongs to Lp(Rn), where 1 < p < ∞,let , where 0 <β < 1, then for the operator TA1 ,A2 ;we have the following estimate,In charpter 5,We consider the multilinear fractionl integral operator defined by:andwherem = m1 + m2and function K (x, y)defmed byK(x,y) and,Pmj+1 (Aj,x,y) defined by:The main results of this charpter are the following ones:Theorems. 1 Let TA,α be the multilinear frctionl operator defined by ( 5.1 ), Then for 0 < r < 1 ,there exists a positive constant C = C(n, r)suchthat for any ,Theorem5. 2 Suppose that Tα,A be the multilinear fractional operator defined by ( 5.1 ), Then for 0 < r < 1 ,there exists a positive constant C depending on n, and such that for any , we have:where g = n/n-α.Theorem S.13 Let TA1,A2,α be the multilinear fractional operator defined by ( 5.3 ), then for 0 < r < 1 ,there exists a positive constant C = C(n,r) such that for any ,Theorem5. 14 Let TA1,A2,α be the multilinear fractional operator defined by( 5.3 ), then for 0 < r < 1 ,there exists a positive constant C depending on n, and , such that for any , λ > 0 , we havewhere q =n/n-α.