Oscillation Multiplier And A Class Of Multi-linear Operator In Function Space,

Since 1950's, singular integrals were established by Calderon and Zygmund. One of the central questions in classical harmonic analysis is the boundedness of operators (such as oscillating multipliers, multilinear operators etc). The bound-edness of operators on various function spaces plays a profound and extensive role, and it developed many methods and techniques have been applied in many fields of mathematics, such as complex analysis, potential theory, operator theory and nonlinear analysis etc. For a general overview of their applications, see [2, 3,6,10,14-17,19,27,32,33,53,54] etc.This thesis focuses on the boundedness of oscillating multiplier operators and multilinear integral operators on the Triebel-Lizorkin spaces, Herz spaces and Herz-type Hardy spaces. Although the Lp-boundedness, Lwp-boundedness, Hp-boundedness of the above operators have been studied thoroughly, no bounded-ness on the Triebel-Lizorkin spaces is obtained. Since the Triebel-Lizorkin space Fpα,q(Rn) provides a uniform setting of many important function spaces in har-monic analysis, such as Lebesgue spaces, Hardy spaces, Sobolev spaces, BMO spaces, Lipschitz spaces etc. On the other hand, we observe that the weight Lp space with power weight|x|ap is the Herz space Kpα,p. Thus, our results can be regarded as the extension of the existed results. Since the Triebel-Lizorkin spaces are more complex than the classical Lebesgue spaces, more additional techniques will be presented in this thesis, which is divided into four chapters.In Chapter 1, we obtain the boundedness of an oscillating multiplier operator TΩ,α,βon the Triebel-Lizorkin space for 0<β< 1. As for different p, q, we partially extend some known results.In Chapter 2, we discuss the boundedness of the operator Tαon the Triebel-Lizorkin space by applying Miyachi's results [48], here Tα= Tα,as the same operator in Chapter 1 forβ= 1. The original motivation is the singularities of the kernel vary for differentβ. Moreover, we also consider the case forβ> 1 and get the same results.In Chapter 3, the boundedness of an oscillating multiplier mr,βfor differentβon the Herz type spaces is obtained, which is based on the results of [39,48]. Ifβis close to 1 or a is suitably large, our results extend the non-weighted case of one of the main results in [39]. Forβ≥1, the results with no weights on the Herz type spaces are also new.In Chapter 4, on the basis of some ideas from [70,72],we weakened the condition of the theorem of [68] and obtained the boundedness of the multilinear fractional integral operator with variable kernel on the Herz-type Hardy and Hardy spaces. Moreover, these results extended some main results of the papers [38,70-72].Chapter 1 Letα> 0,β> 0. Fix a functionΨ∈C(Rn) satisfyingΨ(y)=1 for|y|>1 andΨ(y)=0 for|y|< 1/2. Considering an oscillating multiplier operator TΩ,α,β(f)=KΩ,α,β*f initially demed on f∈S(Rn), where KΩ,α,βis a distribution kernel satisfying andΩis a smooth function on the unit sphere Sn-1 for y≠0.IfΩ≡1, we denote TΩ,α,βas Tα,β. It is very interested in studying os-cillating multiplier operators, since the operator Tα,βis not only related to the convergence of multiple Fourier series, but also related to the Cauchy problems of Schrodinger and Wave equations. If O<β< 1, the operator Tα,βis a typical example of the pseudo-differential operator of the class S1-β,0-α;Ifβ> 1, the oper-ator Tα,βis related to the Cauchy problems of Schrodinger and Wave equations. We look at two examples related to PDE.Example 1. The Cauchy problem of the Schrodinger Equation, has the solutionThe solution u(x,t)is the multiplier operator with multiplierμ(tξ)forα=0 andβ=2.Example 2.The Cauchy problem of the Wave Equation, has the solutionThus,The solutionμ(x,t)is the multiplier operator with multiplierμ(tξ)forα=0 andβ=1.In this chapter,we study Triebel-Lizorkin space estimates for an oscillating multiplier mα,β.At first,we state some known results.S.Wainger[65](1965).Let 0<β<1,1≤p≤∞.(1)If,then(2)If,then is failed. Fefferman-Stein[28](1972).Let 0<β<1.(3)If,thenThus,combining(1),(2)with(3)yield that:Theorem A([28,65]) Let 0<β<1.Ifα=(nβ)/2,then Tα,βis bounded on the Hardy space H1(Rn).Moreover,Tα,βis bounded on the Lebesgue space LP(Rn)if and only if for any fixed p∈(1,∞). It is well known that Lp(Rn)(p≤1) no longer has the same good nature as 1<p<∞. The Hardy space Hp(Rn) is a good substitutes of them when p≤1. For example, the Riesz transforms are not bounded on LP(Rn), but they are bounded on Hardy spaces Hp(Rn). The theory of Hardy spaces plays an important role in the theory of boundedness of operators and partial differential equations, the reader can refer the papers [20,24,42,50,56,59,62,67]. As for Ta,β, what boundedness can be hold on the Hardy space Hp?Sjolin[55](1979). Let 0<β<1,0<p<1. if and only if Moreover, if we use Theorem 4.1 in [46], we have the following result.Theorem B([48,55]) Letβ≠1. The operator Tα,βis bounded on the Hardy space Hp(Rn) if and only if for any fixed p∈(0,∞).The following theorem can be easily deduced by checking the proofs of the previous two theorems.Theorem C(see also [11]) Letβ≠1. The operator TΩ,α,βis bounded on the Hardy space Hp(Rn) if for any fixed p∈(0,∞).Here, we must point that if a andβsatisfy certain conditions, the operator Tα,βis bounded on the L1→H1,Ll→Lq,Hp→L∞,HP→BMO etc, see [48] for details.On the other hand, wavelet analysis becomes very active and useful in both pure and applied mathematics in the last three decades. With the rapidly devel-oping wavelet analysis, one important function space, the Triebel-Lizorkin space Fpα,q(Rn) arises and is well-studied (see [29,30,35-37,63,64,69]). The sig-nificance of the space Fpα,q(Rn) is that it provides a uniform setting of many important function spaces in harmonic analysis, such as Lebesgue spaces, Hardy spaces, Sobolev spaces, BMO spaces, Lipschitz spaces etc. Thus, a natural ques-tion is can we extend Theorems A, B, C to the Triebel-Lizorkin spaces? We answer the question partially in this chapter. Let's give our main results. Theorem 1.1.1 Let 0<β< 1. For any r∈R,1< p≤q≤2 or 2≤q≤p<∞,the operator TΩ,α,βis bounded on the space Fpr,q(Rn) if a≥nβ|1/2-1/p|.Comparing Theorem 1.1.1 with Theorems A and B, we obtain the optimal boundedness of the operators on the spaces Fpr,q(Rn) for suitable p, q. We do not know whether Theorem 1.1.1 is true for all 1< p,q<∞? However, we have the following result.Theorem 1.1.2 Let 0<β< 1. For anyγ∈R,1< p, q<∞or 0< q≤p≤1, the operator TΩ,α.βis bounded on the space Fpγ,q(Rn) if a> nβ|1/2-1/P|.When 0<p≤1<q<∞, we also have the following optimal theorem.Theorem 1.1.3 Let 0<β< 1. For anyγ∈R,0< p≤1< q<(2-β)/(1-β), the operator TΩ,α,βis bounded on the space Fpγ,q(Rn) if a> nβ(l/p-1/2).For q>(2-β)/(1-β), the operator TΩ,α,β, is bounded on the space Fpγ,q(Rn) ifα> nβ(1/p-1/2+σ), whereσ= 1/q'-1/q.Chapter 2 In this chapter, we discuss the boundedness of the operator Tαon the Triebel-Lizorkin spaces, here Ta= Tα,1 as the same operator in Chapter 1 forβ= 1. The original motivation is the singularities of the kernel vary for differentβ. Moreover, we also consider the case forβ> 1 and get the same results.In Chapter 1, we have known that the operator Ta has an intimate connec-tion with the Cauchy problem for the Wave equation. Relying on the properties of the kernel Kα,β, we can see Tαis not a limit case of Tα,βasβ→1, the reason is the range that one expects from Theorem A and B is which con-tracts with the following Theorem E. This essential difference is due to the fact that in the caseβ= 1, the corresponding kernels are singular over|x|= 1, and in the caseβ≠1 the kernels are singular at zero or infinity (about the kernel Kα,β, see [48] for more details). From the above observations, it is meaningful to study Triebel-Lizorkin space estimates for the operator Ta. In 1980. Peral obtained the Lp boundedness of Tαin [52].Theorem E([52]) The operator Ta is bounded on Lp(Rn) if and only if for any fixed p∈(1,∞),Moreover, this result was mentioned by G. Alexopoulos and M. Marias [1. 44] when they studied oscillating multipliers on general Lie groups and manifolds. Later on, it has also been improved and extended by many authors. In 1981, Miyachi extended the LP boundedness of Ta to the Hp space as follows.Theorem F([48]) The operator Ta is bounded on Hp(Rn) if and only if for any fixed p∈(0,∞)Recently, J. Chen, D. Fan and L. Sun [12] obtained the Hardy space esti-mates for the wave equation on compact Lie groups. Meanwhile, D. Fan and L. Sun [25] studied certain kinds of oscillating multipliers Tα,m related to Cauchy problem for the wave equations on Euclidean space and on the torus. As for the endpoint's estimate, the operators Tα,m are bounded from certain block spaces to the Lp spaces. Comparing these results, one is naturally to ask whether the operator Tαis bounded on the Triebel-Lizorkin spaces or not? In this chapter, we partially answer this question.Theorem 2.1.1 Letγ∈R. For any 1< p≤q≤2 or 2≤q≤p<∞, if a≥(n-1)|1/2-1/p|, Then the operator Tαis bounded on the space Fpγ,q(Rn).Theorem 2.1.2 Letγ∈R. For any 1< p, q<∞or O<q≤p≤1, ifα> (n-1)|1/2-1/p|, Then the operator Ta is bounded on the space Fpγ,q(Rn).Theorem 2.1.3 Letγ∈R. For any 0< p≤1< q≤∞, if a≥(n-1)(1/p-1/2). then the operator Ta is bounded on the space Fpγ,q(Rn).Chapter 3 In this chapter, we continue to study the same operator, but the function spaces change to the Herz spaces. In order to avoid confusion caused by symbols, we note the operator as Tγ,βwhose kernel is Kγ,β. The sharp Hp (0< p<∞) estimate for the operator Tγ,βwas well estab-lished by many authors [28,47,48,52,55,65]. Moreover, Chanillo [8] studied the oscillating operators in the weighted LP spaces and showed the following theorem.Theorem G([8]) Let 0<β< 1. If then fails for any fixed p∈(1,∞).As for the Lebesgue space LP with power weights, many people studied the boundedness of some operators on it. In 1957, Stein [58] showed that if T is bounded on Lq(Rn)(1< q<∞) and k(x)< C/(|x|n), then T is bounded on the weighted spacesIn 1994, Soria and Weiss [57] developed Stein's result in the following way. The singular integral operator T can be replaced by any sublinear operator T satisfying the following size condition:for any f∈L1(Rn) with compact support,Moreover, the endpoint case q=1 was supplemented in [57]. It should be pointed out that the above condition is satisfied by many operators in harmonic analysis, such as C-Z operator, Ricci-Stein's oscillatory singular integral, the Bochner-Riesz means at the critical index and so on.The above Stein's result attracts many people to study on the weighted theory of operators. A natural problem is stated as follows:what conditions onωensure that the boundedness on Lωq(Rn) of T can be deduced by the boundedness on Lq(Rn)(1<q<∞)?In 1972, Muckenhoupt first resolved this problem for the Hardy-Littlewood maximal operator M and he obtained that M is bounded on Lωq(Rn) (1< q<∞) if and only ifω∈Aq. Note that |x|βG Aq if and only if -n<β< n(q-1), we can see Stein's result is sharp for power weights.Since then, the weighted theory of operators became a popular direction in harmonic analysis. However, there is another direction to develop Stein's result if one noti'ce that L|x|βq(Rn)=Kqβ/q,q(Rn),where the right side is a special claSS of the homogeneous Herz spaces Kqap(Rn)for-∞<α<∞,0<p≤∞,0<q≤∞. Therefore,the previous question can be developed as to find some conditions on (a,p,q)such that T is bounded on Lq(Rn)implies T is bounded on Kqa,p(Rn).Observe that the weighted Lp space with power weight |x|ap is the trace space Kpa,p(see Chapter 3 for the definition).Thus,by Theorem G,‖f‖Kqa,p(Rn)fails ifα≥n(1-1/q),at least for p=q>1.However,Li and Lu [39]found that the operator T(nβ)/2,βis still bounded from the Herz-type Hardy space HKqα,p to the Herz space Kqαp even in the limit caseα=n(1-1/q)if 0<p≤1<q<∞.More precisely,they proved the following result.Theorem H([39]) Let 0<β<1.If 0<p≤1<q<∞andα= n(1-1/q),thenRemark In[39],the authors actually proved a weighted case.Here,for simplicity,we state the unweighted case.The restrictionα=n(1—1/q)in Theorem H may be so strong,so we want to relax it and get the following result.Theorem 3.1.1 Let 0<β<1,0<p≤1<q≤(2-β)/(1-β) andα≥n(1—1/q). if Then we haveTheorem 3.1.2 Let and Then we haveHere,we write Tγ(f)=Tγ,1(f)for simplicity in notation.Theorem 3.1.3 Letβ>1,0<p≤1<q<∞andα≥n(1—1/q).if Then we haveChapter 4 In this chapter,we consider the boundedness of a class of multilinear fractional integral operators with variable kernel on the Herz-type Hardy and Hardy spaces. Let Sn-1 is the unit sphere in Rn with respect to surface measure dσ(x'). We defineΩ(x,z)∈L∞(Rn)×Lr(Sn-1)(r≥1),if the following conditions are satisfied:Let 0<μ<n,Ω(x,z)∈L∞(Rn)×Lr(Sn-1),then the multilinear fractional integral operator with variable kernel is defined as where Rm(A;x,y)=A(x)-∑1/(?)DγA(y)·(x-y)γ,m≥1,γ=(γ1,γ2,…γn) |γ|<m andγi(i=1,2,…,n)is a nonnegative integer.Writing |γ|=∑i=1nγi,γ!=γ1!γ2!…γn!,xγ=xγ1xγ2…xγn,Dγ=((?)|γ|)/((?)γ1(?)γ2…(?)γn).In 1955,Calderon and Zygmund[4]defined the singular integral operator with variable kernel as and they investigated the Lp boundedness of TΩ.They found that these operators are closely related to the second order linear elliptic equations with variable coefficients.In[4],Calderon and Zygmund decomposedΩas the sphere harmonic functions and they obtained the following result.Theorem I([4]) IfΩ(x,z)∈L∞(Rn)×Lr(Sn-1)and suppose z′=z/|z|,r> 2(n-1)/n,Ω(x,z)satisfying∫Sn-1Ω(x,z′)dσ(z′)=0.then there is a constant C>0 such thatIn 1956,Calderon and Zygmund[5]obtained the boundedness of TΩon more large Lebesgue spaces by using rotation method.Theorem J([5]) Let 1<r<0<∞,1<p<00,p≥r'=r/(r-1).If and satisfying then there is a constant C>0 such that In 1978, Calderon and Zygmund improved the boundedness results of TΩby considering the dimensional of Rn and relaxing the restrictions ofΩ.Theorem K([7]) If 1< r≤2, r> p'(n-1)/n or 2≤r<∞, ThenComparing the two theorems, we found that for fixed p and n, the range of r is bigger in Theorem K.In 1986, Christ and Duoandikoetxea etc [18] further improved Theorem K Namely, if p,γsatisfies then TΩis bounded on Lp(Rn).Analogue to the definition of the singular operator TΩwith variable kernel, we notice that the fractional integral operator TΩ,μwith variable kernel is a natural extension of the fractional integral operator, which is defined asIn 1971. Muckenhoupt and Wheeden [49] obtained the boundedness of power weights of TΩand TΩ,μ. In 2002, Ding, Chen and Fan [21] proved if p≤1, suppose the kernelΩsatisfies certain Lγ- Dini condition, then TΩand TΩ,μ(0<μ< n) are bounded on the Hardy spaces.Later on, Zhang and Chen [70] extended the existed results. They obtained the boundedness of TΩand TΩ,μon the Herz-type Hardy spaces. For more results, see [9,15,38,66,71] etc.It's well known that Herz spaces and Herz-type Hardy spaces are important spaces in harmonic analysis and relevant subjects. Recently, Zhang and Lan [72] studied the boundedness properties of multilinear singular and fractional integrals on the weighted Hardy spaces. As the ideas of the paper [22], combining fractional integral operator with variable kernel with multilinear operator, one is naturally led to the question whether multilinear fractional integral operator with variable kernel is bounded on the Herz-type Hardy and Hardy spaces or not? In this chapter, we discuss this problem. Here,we remark that there are more difficulties to study the operators in this chapter,since they are more complex and can not be considered as a simple extension of commutators.The main tools we used can be seen[22,70,73].In this chapter,we weakened the theorem condition of[68]and obtained the following main results:Theorem 4.1.1 Let 0<μ<n-β,0<β<1,DγA∈∧β.If Q(x,z)∈L∞(Rn)×Lr(sn-1)with satisfies Lr一Dinicondition.Then there exists a constant C>0 independent of f,A such thatTheorem 4.1.2 Let 0<μ<n-β,0<β<1,DγA∈Aβ,n/(n+β)< p<1,1/q=1/p-(μ+β)/n.IfΩ(x,z)∈L∞(Rn)×Lγ(Sn-1)with r>max｛n/(n/(n+β-n/p),n/(n—μ—β)}satisfiesThen there exists a constant C>0 independent of f,A such thatCorollary 4.1 Let 0<μ<n-β0<β<1,0<p1≤p2<oo, D7A∈Aβ. IfΩ(x,z)∈L∞(Rn)×Lr(Rn)with r>max{q1′,q2,n/β}satisfies Lr-Dini condition. Then there exists a constant C>0 independent of f,A such thatRemark 4.1 Suppose where Rmj(Aj；x,y)=Aj(x)-∑|r|<m,1/y1 DrAj(y)·(x-y)r,(j=1,2,…,k),M=∑j=1kmj.Repeating the proofs of the previous theorems,we find these three the-orems above also hold for TA1,A2…,AKΩ,μwith the bounds respectively.