| This PH.D thesis focuses on the weighted boundedness of some fractional integrals。It is well known that the classical fractional integral is defined as follows: For the boundedness of Ia(f),readers can see [74], which is a formal result of fractional integrals. We say a nonegative functionωis belonged to Apclass, wherel< p<∞,if there exists a constant C such that for all cubes Q with sides parallel to the coordinates axes,we have Also we call a nonegative functionωbelongs to A1 class if there exists a constant C such that for all cubes Q with sides parallel to the coordinates axes, we have and we define For more result about Apclass, readers may see [36] and for the weighted boundedness of Iα(f)(x), one can see referece [56].This PH.D Thesis will continue to study the boundedness of commutators of fractional integrals on the basis of the earlier work done by other mathematicians.For the weighted boundedness of Iα(f)(x), readers may see [56] or [57], these results are very famous and attract many attentions by other mathematicians. The main contents of the paper are:1 The boundedness of commutators of vector-valued fractional integrals with the kernel satisfying Hormander type of Young function, where the weight is only a nonegative and locally integrable function.2 The Coifman's estimates and weighted weak LlogL estimates of vector-valued commutators of multilinear fractional integrals. 3 Two-weight norm inequalities for commutators of multilinear potential operators4 Weak type two-weight inequalities for vector-valued commutators of mul-tilinear fractional integrals.This PH.D Thesis contains six chapters.Chapter 1 In this chapter, we give the main function spaces that will be used throughout this paper, such as BMO space, ExpLr space and Orlicz space, and this chapter also contains some useful lemmas and defitions.Chapter 2 In this chapter, we study the boundedness of commutators of vector-valued fractional integrals with the kernel satisfying Hormander type of Young function, including the Coifman's estimates and the weak type LlogL estimates of this operator, where the weight is only a nonegative and locally integrable function. Before we discuss the commutators of fractional integrals, we give the definition of commutators of singular integrals first.Suppose T(f)(x) = p.v.∫Rn K(x-y)f(y)dy is a singular integral operators with its kernel K(x), where K(x) satisfies the following conditions:For more results about T(f)(x), readers may see [74]. If a nonegative and locally integrable function b(x) belongs to BMO(Rn)spaces,(see the definition of BMO in the next section), then we define the following commutators of singular integral operators Tb(f)(x), and its corresponding higher order commutators, Obviously, when m= 1, we have Tb1(f)(x)= Tb(f)(x).In 1976,Coifman, Rochberg and Weiss([19])proved that Tbm(f)(x) is bounded from Lp(Rn)(1< p<∞) to Lp(Rn) if and only if b∈BNO(Rn).Then in 1982,Chanillo([7]) firstly considered the following commutators of fractional operators, Chanillo proved that Iαb(f) is bounded from Lp(Rn) to Lq(Rn) if and only if b∈BMO(Rn),where 1< p<∞and 1/p-1/q=α/n. Now we payattention the fact that both in Coifman or Chanillo's papers, the authors can't give the boundedness of commutators of singular integrals or fractional integrals when p= 1. So the endpoint estimates of commutators of singular or fractional integrals is a considerable questions.This question has not been solved until 1995 by Perez in [58] First he gave a counterexample that the commutators of singular integral is not of weak type(1,1),then he proved that the operator Tb(f)(x;)satisfies a weak type LlogL estimate. His work has been cited by many other mathematicians. In 2001, Ding, Lu and Zhang([29]) also gave a counterexample that Iαb (f)(x) is not of weak type (L1, Ln/n-α), and they proved that Iαb(f)(x) also satisfies a weak LlogL estimates which is adoptted to commutators of fractional integrals. In [29], Ding, Lu and Zhang first introduce a new kind of fractional type sharp maximal function Mα#(f)(x), then by the point estimate of Mα#(Ib(f))(x) and the method in [58], they proved that Iαb(f) also satisfies a weak type LlogL estimates. Then in 2007, Gorosito, Pradolini and Salinas([38]) improved Ding, Lu and Zhang's results to the weighted case by a different method. In their papers, they give the weak type LlogL estimates of Iα(f)(x), then by this results they got the weighted weak endpoint of Iαb(f), the details can be found in [38], and the third chapter of this paper also used this method.In 1972, Coifman ([16])proved the following Coifman's type estimate of T(f)(x):For everyω∈A∞, we have by the above estimate, in 1994, Perez([59])proved that for every locally integrable functionω(x), we have By the above estimates and classical C-Z decomposition, Perez also proved the following weak weighted inequality:also for every locally integrable functionω(x), we have Now a natural question is that for commutators of singular integrals, dothe cor-responding results still hold? These questions were also answered by Perez. In reference [60] and [64], Perez proved the following two inequalities, and (0.13) whereφ>m(t)=t(1+log+t)m and the definition of ML(logL)m+δω(x) will be given in the next section.Now we should pay attention to the fact that the above estimates need the condition that the kernel K(x) should satisfy some smoothness, for example, K(x) should satisfy (0.4) and (0.5). However,in 2005, Martell, Perez and Trujillo-Gonzalez ([54]) gave a counterexample that if the Kernel K(x) satisfy the H1-Hormander condition,then the Coifman's type of singular integrals will fail. So it is interesting to find a new Hormander condition such that Coifman's type estimates can still hold. In 2005, Lorente, Riveros and Torre([50]) gave a new class of Hormander condition HA which is very closed to the Orlicz function and is between H1-Hormander condition and the classical condition of singular integrals for the kernel K(x). If the kernels K satisfy this kind of H'omander condtion, then the singular integrals can still satisfy a new kind of Coifman's estimate, they proved the following questionTheorem A([50]) Suppose that T(f)(x) is a singular integrals which is bouned on Lp(Rn)(1< p<∞) space, and A, If the kernels K(x) satisfies the following LA-Hormander condition: where CA> 0 and for any x∈Rn, we have R> CA|x|.Then for every 0< p<∞andω∈A∞, there exists a positive constant C such that, Then in 2008, Lorente, Martell, Rivers and Torre([52]) considered the commu-tators of singular integrals Tbk(f)(x) with its kernels K(x) satisfying the LA-Hormander condition. They proved the Coifman's type estimate of Tbk(f)(x) and the weak LlogL estimate and the kernel K should satisfy the following con-dition:Recently, Riveros([68])considered the corresponding generalized fractional integral IKα(f)(x)=∫RnKα{x-y)f(y)dy. She proved that if the kernel Ka satisfies the following fractional type Hormander condition, then the corresponding results of (0.9) and (0.10) can still hold for Iα(f)(x). However, for the commutator of the generalized fractional integrals, can we get the corresponding results? In chapter 2, we give a positive answer to this question and the operator we consider is as following, where dy and b∈BMO(Rn).We say a Kernel Kα∈HA,k,αif Kαsatisfy the following condition: (0.20) when k= 0, we denote Kα∈KA,α. Now we give the main results of chapter 2.Theorem 0.1 For 0< p<∞, b∈BMO,ω∈A∞and A, B, Ck be Young functions such that A-1(t)B-1(t)Ck-1(t)≤t with Ck-1(t)=e1/k. If the kernel Kα∈HA,k,α∩HB,α, then there exists a constant C such thatBy Theorem 0.1 and a classical duality argument, we can get the following theoremTheorem 0.2 Let A be a young function. Suppose that there exist Young functionsξandθsuch thatξ∈Bp' andξ-1(t)θ-1(t)≤A-1(t) and set D(t)=θ(t1/P).1) If Iα,b,qk{f)(x) be a linear function and its adjoint Iα,b,q*k(f)(x) satisfies for all 0< q<∞and everyω∈A∞. Then for any nongenative weightμwhich is only locally integrable on Rn, we have for 1< p<∞.Theorem 0.3 Let Iα,b,q(f) and its kernel Ka be as in Theorem 1.1 and Theorem 1.2, suppose that there exists a Young function D satisfying for any locally integrable functionω, then there exists a constant C such that (0.25) whereφk(x)= x[log(e+x)]k for k∈Z+.Remark 0.4 As far as we know, our results are still new even in the unvector-valued case and what's more, our results coincide with the results in [68], so Theorem 1.1-1.3 improve the known results in [68].Remark 0.5 As an application of our main results in this paper, we get the weighted boundedness of commutators of vector-valued fractional integrals with rough kernels. Also, we get the weighted boundedness of commutators of vector-valued fractional integrals associated to a multiplier and we will discuss these facts in Section 7 of this paper.Remark 0.6 In [4], the authors got the endpoint weighted estimates of com-mutators of fractional integrals with the weightωis a only integrable function. However, in [4], the fractional integral they considered has no kernels asΩ= 1, so Theorem 1.3 in our paper can be regarded as a improvement of Theorem 1.5 in [4].Chapter 3 In this chapter, we discuss the Coifman's estimates and weighted weak LlogL estimates of vector-valued commutators of multilinear fractional in-tegrals.In the past ten years, the theory of multilinear operators have also been developed a lot. In 1999年, Kenig and Stein([43])considered the following mul-tilinear fractional integrals, let m∈N, 1/s= 1 /ti+1/t2+…1/tm-α/n> 0, where 0<α< mn,1< ti<∞, and the multilinear fractional integral operators is defined as follows,they proved for some i, if we have ti= 1, then If for every i, we have ti> 1, then In 2002, Grafakos and Torres([39] and [40])comsidered a complete multilinear theory, they proved the multilinear C-Z and its maximal operators are bounded from Lp1 x LPm to LP spaces, where multilinear singular integrals is defined as follows, where the kernel K(x,yi,…, ym)satisfies some certain conditions see reference [39] for details. But for the weighted boundedness of T(f)(x) was not solved until 2009, in [49], Lerner defined a new multilinear weight Ap, and these class of weights are very adopt to the multilinear operators, they proveed the weighted boundednes of T(f)(x) with this new kind of weight. So the next three section of this paper are the weighted boundedness of multilinear commutators of fractional integrals. In chapter, we consider the Coifman's type estimates and weak LlogL estimates of vector-valued commutators of multilinear fractional integrals. At the same time, we proved the boundedness of vector-valued multilinear fractional integrals, which is an extention of [43] and our method are very different from theirs. Now we give the main results of chapter 3.Theorem 0.7 Let 0< p<∞,1/m< q<∞,0<α< mn, and 1/q= 1/q1+…+1/qm, bi∈Oscexp(Lri) for ri≥1, i= 1,2,…,l r= min{ri,…,ri} thatTheorem 0.8 Letω∈A∞,q,q1,…,qm,r1,…,γm,γand bi be as in Theorem 1.1,φ(t)=t(1+log+t)1/r,ψ(t)=t1-γ,(1+log+(t-γ))1/r andφ(t)=(t(1+ log+tγ)1/r)1/1-γ',γ=α/mn,then there exists a constant C,such that for anyλ>0, we haveRemark 0.9As far as we know,our results are new even for the Iα,b(f)(x).Remark 0.10 Obviously.Theorem 0.8 improve the known results in [38] and our results can been regarded as a limit case for the results in [77] whenα→0.Remark O.11 Obvioulsy, for the case when l=0,the Coifman type estimate and weak LlogL estimate can be also proved and we will mention them in chapter 3.Remark 0.12 In the proof of the main results in chapter 3,we also get the boundedness of vector-valued multilinear fractional integral on product Lp spaces.We have the following results:Suppose Iα,q(f)(x)be as the vector-valued multilinear fractional integrals mentioned above,and let where 1 0,0<ε< 1, such that for all k∈Z, we have The main result of chapter 4 are the following,Theorem 0.13 Suppose that 0<α< mn,1< p1,…,pm<∞, q is a number that satisfies 1/m< p≤q<∞for some r> 1. If bi∈Oscexp(Lri) for some ri> 1(i= 1,…,m). Denote ifΨ,Φ1,…,Φm are Young functions that satisfy and for some constant c> 0. Moreover, ifΨ(t) andΦi(t) satisfy the following, for Bm(t)=tlog(e+t)m, then the following inequality, holds for all if the pair of weights (u, v) satisfy one of the following conditions: a)q> 1 and b)q≤1 andRemark 0.14 by [53] or [60], if we setΨ(t)= tq(log(1+t))q-1+δandΦi(t)= tpi'(log(1+t))pi'-1+δ, whereδis a positive constant, thenΨ(t) andΦi(t) satisfy conditions (0.33)-(0.34)。Remark 0.15 Obviously, when our results are new for the commutators of classical multilinear fractional integrals proposed by Kenig and Stein [43].Remark 0.16 Recenetly, Lerner et al. [49] considered the following type commutators of multilinear C-Z singular integrals: where (?)= (b1,…,bk)(bi∈BMO(Rn)) and T(?) is a multilinear Calderon-Zygmund operator, see [39] for details. So the corresponding results for the following commutators of multilinear potential type operators: still satifies (0.36) under the same condition (0.33)-(0.34) for (u,(?)). The proof of this result is even easier if we follow the proof of Theorem 1.1.Remark 0.17 In [60], Perez only gave the two-weight norm inequalities for the potential operators in the case q> 1. And in reference [47], Li also gave the two-weight norm inequalities for the commutators of potential operators when q> 1. So our results not only improved their results when q> 1, but also give the new results in the case q≤1. Chapter 5 In this chapter, we discuss the weak type two-weight inequalities for vector-valued commutators of multilinear fractional integrals.In the last chapter, we get the two-weight norm inequalities for commutators of multilinear potential operators. At this time, we need the pair of weights satisfy some strong condition, if we weaken the "power and log bump" of the pair of weights, what will happen? This question has also been answered by Cruz-Uribe and Perez, in 2000, they([18]) gave a positive answer to this question. They proved if the pair of weights (u, v) satisfy some weakened "Power and log bump " condition, then the commutator of singular integral will satisfy two-weight weak (p,p) inequalities, where 1< p<∞. In 2004, Liu and Lu ([48]) also consiered the two-weight weak (p,p) inequalities of commutators of fractional integrals. In the above papers, the author used some maximal operators which is related to Orlicz function to treat the space with C-Z decomposition. Recently, Tang( [77]) proved the weak two-weight norm inequalities for vector-valued multilinear singular integrals, motivated by the above work, we consider the operator Iα,b,q(f) which has been appeared in chapter 2 also satisfies the weak two-weight norm inequalities. The main results of chapter are the following,Theorem 0.18 Let 1< p<∞,1/p= 1/p1+…+1/pm,1/m< q<∞and 1< q1,…,qm. Given pairs of weights (μ, vj) satisfying for some r> 1 and for all cubes Q and Cj(t)= tp'log(e+t)p'j with j= 1,…, m. Then there exists a constant C such thatRemark 0.19Obviously, our results improved the main results in [48].Chapter 6 In this chapter, we briefly discuss that whether the operators with variable kernels can satisfy LlogL estimate and the weighted norm inequal-ities, where the weight is only a locally integrable function. Also in chapter, we will point out some known results about operators with variable kernels, some of them are our recently publication works.
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