Types Of Operator Sector And Related Issues

Harmonic analysis derived in eighteen century, it becomes a very importantbranch in Mathematics after its rapid and deep development. The boundednessof some operators plays a profound and extensive role in harmonic analysis.This thesis will study singular integrals, commutators and pseudo-differentialoperators as well as their associated problems. It is divided into four chapters: InChapter 1, we study the boundedness of certain classes of singular integrals alongsurface on the homogeneous Triebel-Lizorki space; In Chapter 2, we apply thetechnic of the Sharp function and obtain weighted norm inequalities for maximalvector-valued operators and commutators, without any additional condition onthe weight functions; In Chapter 3, we consider operators associated with thesections coming from the Monge-Ampere equation; In Chapter 4, we study certainDeleeuw type theorems on some non-convolution operators from Euclideanspaces to Torus and obtain some useful results on Torus by application. In thefollowing, we will state the main content of every chapter.Chapter 1 Let n≥2 andΩ(y')∈L¹(S^n-1), where S^n-1 is the unite spherein Rⁿ with respect to surface measure da. Throughout this paper, we also assumethatFor f∈Sⁿ, the singular integral operator is defined byIn 1950s last century, the investigation of the operator T began with A.P. Calderon-A. Zygmund's pioneering study, see [19]. They [20] introduced themethod of rotations and proved that suppose thatΩ∈L ln⁺ L(S^n-1), then Tmaps L^P(Rⁿ) to itself, 1＜p＜∞. We also remark that T extends to an operator bonded from L²(Rⁿ) into itself if and only ifBy simple calculation, the conditionΩ∈L ln⁺ L(S^n-1) implies the condition(0.0.3). In 1979, Ricci and Weiss [61] gave a characterization of the Hardy spaceH¹(S^n-1) and used the method called C-Z Rotation to prove that the conditionΩ∈H¹(S^n-1) implies that the L^P(Rⁿ)-boundedness for T, 1＜p＜∞, also see[14] for details. We point out that for r＞1,and all inclusions are proper, so C-Z's result is improved. In 1998, Grafakos andStefanov [36] found another sufficient condition onΩ, that is, for someβ＞1For convenience, we denote by F_β(S^n-1), whose element satisfies both (0.0.1)and (0.0.4). Using Holder's inequality, it is easy to check that forβ₁＞β₂＞0, F_β1(?)F_β2. And for any r＞1, L^r(S^n-1)(?)F_β(S^n-1). We denote by∩_β＞1F_β(S^n-1) = F_∞(S^n-1). Grafakos and Stefanov [36] also gave examplesto show that F_∞(S^n-1)(?)H¹(S^n-1)(?)F_∞(S^n-1). They apply the idea ofLittlewood-Paley decomposition in [26] to prove that ifΩ∈F_β(S^n-1),β＞1,then T is bounded on L^p(Rⁿ),(?)＜p＜β+1. In face, the range of p canextend to (?)＜p＜2β, see [28] for details. But for the endpoint case, it isunclear until now.In this chapter, we shall study two general claasses of singular integralsprovideΩ∈F_β(S^n-1),β＞1. One is defined bywhere b∈L^∞(R⁺), andΓis a suitable C¹ function such that (0.0.5) exists. IfT(t) = t, we set T_Ω,Γ,b=T_Ω,b. And if b(t) = 1, we setΓ(t)=t,T_Ω,Γ. Furthermoreif we assumeΓ(t) = t, then T_Ω,b is reduced to the well-known singular integral operator T which is defined above. Qassem in [4] proved the L^p-boundedness forthe operator T_Ω,Γ given by (0.0.5).Theorem A LetΩ∈F_β(S^n-1) for someβ＞1 and p∈((?),2β). ThenT_Ω,Γ is bounded on L^p(Rⁿ) ifΓis an nonnegative and strictly monotonic functionsatisfying(1) |Γ'(t)|≥C|Γ(t)|/t,(2)λΓ(t)≤Γ(2t)≤CΓ(t) ifΓis increasing, andλΓ(2t)≤Γ(t)≤CΓ(2t) ifΓisdecreasing for all t＞0 and some C≤A＞1.Another is defined bywhereγis a suitable C¹ function such that (0.0.6) exists. Pan, Tang and Yangin [59] proved the L^p-boundedness for the singular integral in (0.0.6).Theorem B Letγbe a C¹ function such thatγ(0) =γ'(0) = 0 andγ' beconvex and increasing. IfΩ∈F_β(S^n-1) for someβ＞1, then T_Ω,γ is bounded onL^p(Rⁿ⁺¹) with p∈((?),2β).Soon, Cheng and Pan [23] obtained the same result as Theorem B forγbeingpolynomials.Theorem C LetΩ∈F_β(S^n-1) for someβ＞1. Supposeγis a polynomialwithγ'(0) = 0. Then for p∈((?),2β),where C is independent of the coefficients ofγ. And when n=2, the conditionγ'(0) = 0 can be removed.There are two aims in this chapter. One is to establish the (?)-boundednessof the operators given by (0.0.5) and (0.0.6). Another is that we consider another hypotheses onγassociated with the definition as follows.Definition 1.1.1 We say that a C¹ functionφ(t) defined on R⁺ satisfies ConditionD ifφ(t) is a positive (or negative) and strictly monotonic function satisfying(â… ) |φ'(t)|≥C|φ(y)|/t;(â…¡)φ(2t)≤C_φ(t), ifφis increasing,φ(t)≤C_φ(2t), ifφis decreasing for allt＞0.It is noted that the operators we consider in this chapter are not convolutionoperators in the usual sense. In this chapter, we view both of them as the sum ofa sequence of convolution measures. Using the ideas of [26] and [38], we obtaina useful theorem and apply it to get the main theorems in this chapter. DenoteF_∞(S^n-1)=∩_β＞1F_β(S^n-1). The mains theorems are as follows:Theorem 1.1.1 Let 1＜p,q＜∞,α∈R andΩ∈F_∞(S^n-1). Then T_Ω,Γextends to an operator bonded from (?)(Rⁿ) into itself in either of the followingsituations,(â… ) IfΓis a C¹, convex and increasing function withΓ(0) = 0;(â…¡) IfΓis as in Theorem A.Theorem 1.1.2 Let 1＜p,q＜∞,α∈R andΩ∈F_∞(S^n-1). Then T_Ω,γextends to an operator bounded from (?)(Rⁿ⁺¹) to into itself in either of thefollowing situations,(â… ) If bothγandγ' satisfy Condition D. Moreover,γ" is monotonic;(â…¡) Ifγis as in Theorem B or Theorem C.If we set the radial function b(|y|)=e^iγ(|y|), then the operator T_Ω,b is just theoscillatory integral considered in [62]. We have the following theorem.Theorem 1.1.3 LetΩ∈F_∞(S^n-1). Ifγ' satisfy Condition D andγ" is monotonic.Then T_Ω,Γ extends to an operator bonded from (?)(Rⁿ) into itself,1＜p,q＜∞,α∈R. Remark 0.0.1 Whenα= 0, q = 2 and 1＜p＜∞,(?)(Rⁿ)=L^p. LetΩ∈F_∞(S^n-1), our results extend Theorem A, Theorem B and Theorem D. Inaddition, our result in Theorem 1.1.3 is new even in the Lebesgue space.Chapter 2 Let k be a function on Rⁿ×Rⁿ\â–³(â–³={(x,x),x∈Rⁿ}). Wesay that k is a standard kernel ifå’ŒLet T be a Calderon-Zygmund operator T with a standard kernel. The correspondingmaximal operator is defined bywhere k_ε(x)=k(x)_{χ{|x|＞ε}}(x) and f is any bounded function with compact support.It is well known that the operators T is of strong-type (p, p), 1＜p＜∞and of weak-type (1,1) (also see [25], P99). In this chapter, we only consider thecase k(x,y)=k(x-y) andδ=1.In 1976, Coifman, Rochberg and Weiss [17] use the commutator to give acharacterization of the BMO space. Generally, for appropriate functions f andb by T_b^m, m＞1where the operator T_b¹f(x)=b(Tf)(x)-T(bf)(x). Unlike the classic theory ofsingular integral operators, Perez [52] gave simple examples which showed thatthe commutator T_b¹ fails to be of weak-type (1,1) when b∈BMO, the space offunctions having bounded mean oscillation. In fact, Perez [52] use the technicof the sharp function to obtain a L log L-type estimate: for any A＞0, |{x∈Rⁿ:|T_b¹f(x)|＞λ}|≤C‖b‖_BMO (?)dx, whereφ(t) = t(1+ log⁺ t). Subsequently, Perez [54] obtained the weighted norm inequality for T_b^m. Andusing the theory of Calderon-Zygmund decomposition, Perez and Pradolini [55]proved the corresponding endpoint estimate.In this chapter, we consider the maximal vector-valued singular integraloperator on {f_j}_j=1^∞defined byand the maximal vector-valued commutator on {f_j}_j=1^∞defined bywhere f = {f_j}_j=1^∞is any sequence of bounded functions with compact support, b_iis locally integrable. In particular, when f_i = 0, i≥2, the operator T_q^* is just themaximal Calderon-Zygmund singular integral T^*. When b₁=b₂=…=b_m=b,f_i≡0,i≥2, [(?),T]_q^* is the maximal operator of T_b^m. When m = 1, T_b^m is denotedby T_b^*. In [46], Li, Hu and Shi obtain weighted norm inequalities of T^* and T_b^* bythe technic of the Sharp function. We remark that the operators we consider inthis chapter are not linear, and their forms are complex. In this chapter, basingon [55], [56] and [57], we also apply the idea of [46] and obtain the followingresults.Theorem 2.1.1 Let 1＜p,q＜∞,δ＞0 and b_k∈BMO(Rⁿ),1≤k≤m.Then there exists a positive positive constant C such that for any weight functionw, we haveandwith any sequence of bounded functions {f_j}_j=1^∞with compact support, where‖(?)‖=(?)‖b_k‖_BMO. Theorem 2.1.2 Suppose that 1＜q＜∞,δ＞0 and b_k∈BMO(Rⁿ),1≤k≤m. Then there exists a positive constant C such that for eachλ＞0and any weight function w,andwith any sequence of bounded functions {f_j}_j=1^∞with compact support, whereφ_m(t)=t(1+log⁺ t)^mRemark 0.0.2 When m = 1 and f_i≡0, i≥2, our results coincide withthose in [46].Chapter 3 Let a family of open and bounded convex F={S(x,t):x∈Rⁿ,t＞0} be sections. We defineIt is easy to check that d is a quasi-distanc, see Section 1 in this chapter. Inaddition, we also assume that a positive Radon measureμsatisfying thatμ(Rⁿ)=+∞, and satisfies the following doubling property with respect to the parameter,with any S(x,t)∈F. We study the analysis on the space of (Rⁿ, d,μ).Caffarelli and Gutierrez [10] defined the singular integral operator H withthe kernel k(x,y) associated with the sections formally asand they proved that H is of type (2,2). Then, Incognito [42] proved that the kernelk(x,y) satisfies the H(?)rmander condition, and by applying Calderon-Zgmund decomposition on homogeneous spaces, Incognito obtained that H is of weak-type(1,1). In fact, H is also of type (p,p), 1＜q＜∞by interpolationand duality. Recently, for b∈BMO, the space of functions having boundedmean oscillation, Tang [68] considered the commutator of H which is given byH_b f=b(Hf)-H(bf) and obtained the following weighted estimates.Theorem D If b∈BMO and w∈A₁, then for any bounded function f withcompact support, the following inequalitiesandhold, whereφ(t)=t(1+log⁺ t).In this chapter, we consider maximal operators. The corresponding maximaloperator is defined byand the corresponding maximal commutator is defined byIt was proved that H^* is of weak-type (1,1) in [41]. In this chapter, we studyweighted endpoint estimates for H^* and H_b^* defined above. Alphonse [3] studiedthe maximal commutator of the Calderon-Zygmund singular integral operatorwith a standard kernel. However, his method is useless in our cases. In thischapter, improving the estimate of the kernel function k, we get a Coltor typeinequality and prove the following theorems.Theorem 3.1.1 Let b∈Osc_{exp L^q},q≥1 and w∈A₁。For each boundedfunction f with compact support, it holds that where Osc_{exp L^q} is another space of functions having bounded mean oscillation,the case q = 1 corresponds to the BMO space.Theorem 3.1.2 Suppose that b∈Osc_{exp L^q},q≥1 and w∈A₁. Then forany bounded function / with compact support, it holds thatwhereφ(t)=t(1+log⁺ t)^1/q.Remark 0.0.3 We observe that by letting q be as close as to∞, we get asclose as we want to as the L¹ norm in the right hand. Recalling that the spaceOsc_{exp L¹} corresponds to the BMO space, our result in Theorem 1.2 extends thatof Theorem A.Remark 0.0.4 Our results extend those of Theorem D.Chapter 4 Let A is an L^∞function on Rⁿ. Forε＞0, the multiplier operatorT_ε, with symbolsλ(εξ) is initially defined on f∈S(Rⁿ) byAnalogously, we define another multiplier operatoron any g∈C^∞(Tⁿ), where (?) a_ke^{2Ï€ix·k} is the Fourier series of g, and n-torusTⁿ can be identified with Rⁿ/(?), where (?) is the unit lattice which is an additivegroup of points in Rⁿ having integer coordinates. Again, we denote T_ε= T ifε= 1. The relation of the L^p boundedness between T and (?) can be brieflyconcluded by a theorem of Deleeuw [24], which says thatTheorem E Letλis a continuous function on L^∞(Rⁿ), 1≤p≤∞. The following conclusions are equivalent.(â… ) T is bounded on L^p(Rⁿ);(â…¡) (?) is uniformly bounded on L^p(Tⁿ) forε＞0.In 1993, Liu and Lu [47] extended the DeLeeuw theorem to H^p (the Hardyspaces), 0＜p≤1. On the other hand, Fan and Sato [31] extended the DeLeeuwtheorem to the multilinear multiplier operators and obtained on the torus a analogof a famous theorem of Lacey and Thiele [48] about the bilinear Hilberttransform. This theorem, as well as a theorem on the corresponding maximaloperator were later extended to many different function spaces, see [1], [27], [9],[44] and [45].we notice that multipliers are convolution operators while many importantoperators in harmonic analysis are not convolution. Among these operators, twoimportant classes are the commutator defined byand the pseudo-differential operator defined bywith bounded functions f with compact supports, where b∈BMO, m(x,·)∈L^∞(Rⁿ), and T_εis an bounded operator on L^p(Rⁿ) defined above. In recent years,fruitful results of these two operators on Rⁿ were obtained, see [21], [39] and [12].Thus, a natural question is if we can obtain certain DeLeeuw type theoremson these operators by transferring some known boundedness results from Rnto obtain boundedness of their corresponding operators in the n-torus. Withthis motivation, we obtain certain Deleeuw type theorems on these operators bytransferring some known boundedness results from Rⁿ to obtain boundedness oftheir corresponding operators on the n-torus Tⁿ. The main results in this papersare the following:Theorem 4.1.1 Let 1≤p≤∞,λ∈L^∞∩C(Rⁿ) andε＞0. Suppose that T_εisbounded on L^p(Rⁿ). Moreover, for all f∈L^p and b∈BMO, it holds that Then,for all g∈C^∞(Tⁿ), where (?)(g)(x) = b(x)(?)(g)(x)-(?)(gb)(x), and b is anyperiodic BMO function.Remark 0.0.5 We extend the DeLeeuw type theorem to two classes of generalcommutators, see Section 2 in this chapter for details.Theorem 4.1.2 Let 1≤p≤∞, and m(x,·)∈L^∞∩C(Rⁿ) uniformly onx. If P is bounded on L^p(Rⁿ), then for any g∈C^∞(Tⁿ), we havewhere P(g)(x)=(?)m(x,k)a_ke^{2Ï€ix·k}, and m(x,ξ) is periodic on the x-variable.Remark 0.0.6 We extend Theorem 4.1.2 to its commutator, see Section 3 inthis chapter for details.Recalling the Bochner-Reisz operator ofαorder B_ε^α, it is a multiplier with symbolm_α(ξ)=(1-|εξ|²)₊^α. Its cmmutator is defined by B_b,ε^α(f)(x)=b(x)B_ε^α(f)(x)-B_ε^α(fb)(x). For B_b,ε^αby our transference result and results in [39], we have thefollowing theorems.Theorem 4.4.1 Suppose that 0＜α＜1/2 and (?)＜p＜(?). Thenthere exists a positive constant C such thatwhere (?)(g)(x) = b(x)(?)(g){x) - (?)(gb){x), b is any periodic BMO function.Theorem 4.4.2 Suppose that n≥3,(?)＜α＜(?) and (?)＜p＜(?).Then there exists a positive constant C such that where (?) is defined as in Theorem 4.4.1, and b is any periodic BMO function.Similarly, by applying Theorem 4.1.2 and its general one, we transfer some knownboundedness results in [21], [7] and [12] from Rⁿ to obtain boundedness of theircorresponding operators on Tⁿ, see Section 4 in this chapter for details.