Types Of Operators On The Product Space Bounded

Since the fifties the twentieth century, that is Calderon and Zygumund [7]established the theory of singular integral operator (C-Z operator), one of thecentral questions in classical harmonic analysis is the research regarding theboundedness of singular integral operators on various function spaces. This PH.D thesis focuses on this problem. We mainly study the boundedness of somesingular integrals on product spaces. There are five chapters in this thesis.Chapter 1 of this thesis consists of two sections. In section 1, a review of someknown definitions and properties regarding Muckenhoupt weights, Duoandikoetxearadial weights and Muckenhoupt weights on Rⁿ×R^m. We also introduce a newradial weights on Rⁿ×R^m, and further study its properties. Some fundamentalfunction spaces which are needed in the following chapters are given in section 2.In Chapter 2, we mainly discuss the boundedness on product Triebel-Lizorkinspaces of the rough hyper singular integrals T_{Ω,α,β}(α,β≥0), the fractional integraloperators and the Littlewood-Paley funtions.First let S^N-1(N = n or m) be the unit sphere in R^N(N≥2), with normalizedLebesgue measure dσ=dσ(·). For nonzero point z∈R^N, we definez' = (?). The rough hyper singular integrals T_Ω,α and the maximal operator T_Ω,α^*are defined byfor all functions f∈S(Rⁿ), where b∈L^∞(R₊¹),α≥0,Ω∈L¹(S^n-1) is ahomogeneous function of degree zero and satisfieson all spherical polynomials Y_k(y') with degrees k∈[α]. In 1969, Wheeden [90] first obtained that if 0＜α＜2, b≡1,Ω∈L¹(S^n-1)∩(0.0.3), T_Ω,α is the boundedness of ((?),L^p) and weak ((?),L¹), where (?) is thehomogeneous Sobolev space. In 2003, Chen, Fan and Ying [19] considered T_Ω,α,T_Ω,l^*(l is an integer) forα≥0 and proved thatTheorem 0.0.1 Let 1＜p＜∞, (?)=max{p,p/(p-1)}. Suppose thatΩ∈H^r(S^n-1) with r=(?) and satisfies (0.0.3) for all Y_k whose degrees k≤Nwith 2(N + 1)＞α(?). Then there exists a constant C＞0 independent of f suchthat.Later on, in [11, 25], they all dropped the restriction thatα=l for T_Ω,β^* andimproved the cancellation onΩin Theorem 0.0.1. They proved that in Theorem0.0.1, one only needs to assume thatΩsatisfies the cancellation condition (0.0.3)for all k≤[a]. In [24], the author also proved the weighted boundedness. At thesame time, in [18], Chen, Fan and Ying further studied the boundedness propertiesof T_Ω,α (α＞0) on Triebel-Lizorkin spaces and established the following:Theorem 0.0.2 Let 1＜q,p＜∞.(?)= max{p,p/(p-1)),(?)=max{q,q/(q-1)},andα＞0. Suppose thatΩ∈H^r(S^n-1) with r = (n -1)/(n -1+α) and satisfies(0.0.3) for all Y_k of degrees k≤N and 4(N + 1)＞α(?). Then there exists aconstant C＞0 independent of f such thatThe rough hyper singular integral operators T_{Ω,α,β}(α,β≥0) on productspaces are defined byf are test functions in S(Rⁿ×R^m), b∈L^∞(R₊¹Ã—R₊¹) andΩ∈L¹(S^n-1×S^m-1)satisfies whereγ₁,γ₂ are multiple indices, and K and J are some integers. Especially,K = 0 (J = 0) whenα=0(β=0).We denote T_{Ω,α,β} by T_Ωifα=β=0. In 1982, by using the square functionmethod, Fefferman and Stein [51] proved that if b≡1,Ωsatisfies certainsmoothness and cancellation conditions, T_Ωis bounded on L^p(Rⁿ×R^m), where1＜p＜∞. In 1986, by Fourier estimates and Littlewood-Paley theory whichis introduced in [43], Duoandikoetea and Rubio De Francia obtained the aboveresult by assuming that b∈△₂,Ω∈L^r(S^n-1×S^m-1)∩(0.0.5), r＞1. In2002, by using rotation method, Chen [13] weakened the condition onΩ, thatisΩ∈L(log⁺L)²(S^n-1×S^m-1). In 2006, Al-Salman etal [4] also proved theconclusion with different method. Wang [86] improved the result in [13] to thehomogeneous product Triebel-Lizorkin spaces (?). For the caseα,β≥0,Chen, in her dissertation [24], obtained the following:Theorem 0.0.3 Let 1＜p＜∞, (?)=max{p,p/(p-1)},b∈L^∞(R₊¹Ã—R₊¹). SupposethatΩ∈L(log⁺L)²(S^n-1×S^m-1) and satisfies (0.0.3), where 4(K + 1)＞α(?), 4(J+1)＞β(?). Then there exists a constant C＞0 independent of / suchthatIn this chapter, we also study the other two classes operators. Let 0＜α＜n, 0＜β＜m,Ω∈L¹(S^n-1×S^m-1). The rough fractional integral operatorF_{Ω,α,β} on product spaces is defined on all f∈S(Rⁿ×R^m) byForσ∈L¹(Rⁿ×R^m), we defineσ_s,t(x,y)=2^-sn-tmσ(?). The Fouriertransforms ofσ_s,t denotes (?)(ξ,η)=(?)(2^sξ,2^tη). The Littlewood-Paley g functiong(f) on Rⁿ×R^m is defined on f∈S(Rⁿ×R^m) bywhere F_s,t(f)(x,y)=σ_s,t*f(x,y). For any real numbersα,β, we defineIn [18], the authors also studied the boundedness of the above two classesoperators (one-variable) on the homogeneous Triebel-Lizorkin spaces. In thischapter, by Fourier estimates and Littlewood-Paley theory, we extend the boundednessof singular integral operator T_{Ω,α,β} to Triebel-Lizorkin spaces. Also, usingideas in [12, 96], we weaken the cancellation conditions onΩin [24]. We summarizeour result below.Theorem 0.0.4 Let 1＜q,p＜∞, let b∈L^∞(R₊¹Ã—R₊¹) and (?)=(α_o,β_o)∈R×R, (?)=(α_o+α,β_o+β). Suppose thatΩsatisfies (1.6), with K≥[α] andJ≥[β]. AssumeThen there exists a constant C＞0 independent of f such thatTheorem 0.0.5 For 1＜q,p＜∞. Let (?)=max{p,p/(p-1)}, (?)=max{q,q/(q-1)}, let (?)= (α_o,αβ_o)∈R×R, (?) = (α_o-α,β_o-β). Suppose thatΩ∈L^r(S^n-1×S^m-1), r＞1. If 0＜α,β＜(?), then there exists a constant C＞0 independentof f such thatTheorem 0.0.6 For 1＜q,p＜∞. Let (?)=max{p,p/(p-1)}, (?)=max{q,q/(q-1)}, let (?)=(α,β)∈R×R. Ifα∈(-μ₂,μ₁),β∈(-v₂,v₁) and satisfy -(?)＜α＜(?),-(?)＜β＜(?). Suppose thatσsatisfies the following:(i)‖(?)|σ_s,t|*f‖_{L^p(Rⁿ×R^m)}≤C‖f‖_{L^p(Rⁿ×R^m)},(?) f∈S(Rⁿ×R^m),1＜p＜∞,(ii)|(?)(ξ,η)|≤C min{|ξ|^μ₁|η|^v₁,|ξ|^μ₁|η|^-v₂,|ξ|^-μ₂|η|^v₁,|ξ|^-μ₂|η|^<sup>-v₂},for someμ_i,v_i＞0, i = 1, 2. Then there exists a constant C＞0 independent of f such thatNext, we give an application of Theorem 0.0.6. Let B(u, v) be supported in[0,1]² satisfiesDenoteandwhereα,β∈R,Ω∈L～1(S^n-1×S^m-1) satisfies (0.0.5).Therefore, we easily obtain the corollaries of Theorem 0.0.6 as follows.Corollary 0.0.1 Let (?) be same as in Theorem 0.0.6. Suppose thatΩ∈L^r(S^n-1×S^m-1),r＞1 satisfies (0.0.5). Ifα,β∈(?), then we havewhere C＞0 is a constant independent of f.Especially, let B(u, v) = b(2^su,2^tv)_χI(u,v), where I = [0,1]². Let M_s,t(f)(x,y)=σ_s,t*f(x,y), Thenis the well-known Marcinkiewicz integral on product spaces. For any real numbersα,β, we defineCorollary 0.0.2 With the same conditions of Corollary 0.0.1, by assuming thatb satisfies (0.0.9), we havewhere C＞0 is a constant independent of f. Chapter three in this thesis is devoted to the boundedness of the roughmaximal hyper singular integrals T_{Ω,α,β}^*(α,β≥0) on product spaces. We defineaswhere f is a test function in S(Rⁿ×R^m),α,β≥0, b₁,b₂∈L^∞(R₊¹) andΩ∈L¹(S^n-1×S^m-1) satisfies (0.0.5).We denote T_{Ω,α,β}^* by T_Ω^* , ifα=β=0. In 1982, by using the square functionmethod, Fefferman and Stein [51] proved that if b₁≡b₂≡1,Ωsatisfies certainsmoothness and cancellation conditions, T_Ω^* is bounded on L^p(Rⁿ×R^m), where1＜p＜∞. In 1988, by using rotation method, Krug [60] discovered that ifb₁≡b₂≡1,Ω∈L¹(S^n-1×S^m-1), andΩ(-x',y')=-Ω(x',y')=Ω(x',y'),then T_Ω^* is bounded on L^p. In 2002, Wang, in her dissertation [86], also obtainedthe result, by assuming that b₁,b₂∈L^∞,Ω∈L^r(S^n-1×S^m-1)∩(0.0.5), r＞1,and b₁≡b₂≡1,Ω∈GS₃^*(γ)∩(0.0.5),γ＞0, for p∈(?).T_Ω^*is bounded on L^p. In 2006, Al-Salman, Al-Qassem and Pan [4] weakened thecondition onΩ, that isΩ∈L(log⁺L)²(S^n-1×S^m-1).In this chapter, we extend the above results, our main theorem is as follows.Theorem 0.0.7 Let 1＜p＜∞,α,β≥0, b₁,b₂∈L^∞(R₊¹). Suppose thatΩsatisfies (0.0.5) with K≥[α] and J≥[β]. AssumeThen we havewhere C＞0 is a constant independent of f.In Chapter 4, we considered the weighted boundedness of a class generalMarcinkiewicz integrals on product spaces. In 1958, Stein [76] first introducedthe following Marcinkiewicz integralμ_Ωof higher dimension, whereΩis a homogeneous function of degree zero withΩ∈L¹(S^n-1) and satisfiesHe also studied its boundedness on L^p. Later on, a lot of works were doneon the problem. In 1990, Torchinsky and Wang [81] proved that ifΩ∈Lip_γ,0＜γ≤1 and b≡1, thenμ_Ωis bounded on L^p(w) for 1＜p＜∞and w∈A_p (theMuckenhoupt weight class). In 1998, Sato [75] weakened the condition onΩ, thatisΩ∈L^∞(S^n-1). In 1999, Ding, Fan and Pan [38] proved that ifΩ∈L^r(S^n-1),r＞1 and b∈L^∞(R₊¹), thenμ_Ωis bounded on L^p(w) in one of the followingconditions: r'＜p＜∞and w∈A_p/r'; 1＜p＜r and w^1-p'∈A_p'/r'; 1＜p＜∞and w^r'∈A_p. In 2002, Duoandikoetea and Seijo [44] obtained the same resultwith different method. In 2004, the authors [62] showed thatμ_Ωis the weightedL^p (1＜p＜∞) boundedness for certain radial weights A_p^I(Rⁿ)∩RA_p(Rⁿ) whichare introduced by Duoandikoetxea [42], whenΩ∈H¹(S^n-1) and b∈L^∞(R₊¹).In 2008, Zhang [102] introduced a new weight class (?)(Rⁿ)(RA_p(?)A_p^I)and proved that A_p^I∩RA_p can be replaced by (?) in above result.In this chapter, we consider a class general Marcinkiewicz integralsμ_{Ω,α,β}(α,β≥0) defined as follows:whereb∈L^∞(R₊¹Ã—R₊¹),Ω∈L¹(S^n-1×S^m-1) and satisfies (0.0.5).Ifα=β=0, we denoteμ_{Ω,α,β} byμ_Ω, this is the well-known Marcinkiewiczintegral on product spaces. In 2000, Chen, Ding and Fan [14] proved that ifΩ∈L^r(S^n-1×S^m-1)∩(0.0.5), r＞1, thenμ_Ωis boundedness on L^p (1＜p＜∞).In 2001, Chen etal [17] weakened the kernel condition toΩ∈L(log⁺L)²(S^n-1× S^m-1). In 2002, in [18], they further improved the condition onΩ. In 2005,Al-Salman etal [3] and Li [65] all obtained the result, by assuming thatΩ∈L(log⁺L)(S^n-1×S^m-1)∩(0.0.5). There are the other results, see [1, 64, 100].We can refer the papers [?, 91, 58] for the boundedness of the operatorμ_Ω,α(a≥0) (one-variable) on the homogeneous Sobolev spaces. In 2005, Jiang [56]studiedμ_{Ω,α,β} forα,β≥0 and proved the following theorem.Theorem 0.0.8 Let 1＜p＜∞, (?)= max{p,p/(p-1)}. Suppose thatΩ∈L(log⁺L)²(S^n-1×S^m-1)∩(0.0.5) with K＞[(?)-1],J＞[(?)-1]. Then thereexists a constant C＞0 independent of f such thatIn this chapter, by using Fourier estimates and Littlewood-Paley theory,applying the properties of the weights (?)(Rⁿ×R^m), we obtain the weighted L^pboundedness ofμ_{Ω,α,β} for w∈A_p(Rⁿ×R^m), 1＜p＜∞. Also, using ideas in[12, 96], we weaken the cancellation conditions onΩin [56]. We summarize ourresult below.Theorem 0.0.9 Let 1＜p＜∞, w∈(?)(Rⁿ×R^m). Letα,β≥0,b∈L^∞(R₊¹Ã—R₊¹). Suppose thatΩsatisfies (0.0.5) with K≥[α] and J≥[β].AssumeΩ∈L¹(S^n-1×S^m-1) in the caseα,β＞0;Ω∈L(log⁺L)(S^n-1×S^m-1) in the caseαβ=0 andα+β＞0;Ω∈L(log⁺L)²(S^n-1×S^m-1) in the caseα=β=0.Then we havewhere C＞0 is a constant independent of f.In the last chapter, we mainly discuss the parameterized Marcinkiewicz integralswith variable kernels. First, give some definitions. A functionΩ(x, y)defined on Rⁿ×R^m is said to be in L^∞(Rⁿ)×L^r(S^n-1),r≥1, ifΩsatisfies thefollowing conditions: 1.Ω(x,λy)=Ω(x,y), for any x,y∈Rⁿ andλ＞0,2.‖Ω‖₍L^∞(Rⁿ)×L^r(S^n-1))=sup_x∈Rⁿ((?)(x,y')|^r dσ(y'))^1/r＜∞,where y'=y/|y| for any y∈Rⁿ\{0}.Ωis said to satisfy the cancellationcondition, ifThe singular operator T_Ωwith variable kernel byIn 1948, Mihlin [69] first defined and studied this operator (see also [69]). In1955, Calderon and Zygmund [8] proved the L2 boundedness of T_Ω. In 1978, theyfurther proved the L^p boundedness. People also found that these operators canbe applied to the second order linear elliptic equations with variable coefficients.In this chapter, we will study the parameterized Marcinkiewicz integral withvariable kernelsμ_Ω^ρdefined byIfρ= 1, we denoteμ_Ω^ρbyμ_Ω, which is the Marcinkiewicz integrals withvariable kernels. In 2004, Ding, Lin and Shao [40] proved thatμ_Ωis a boundedoperator of type (2,2), ifΩ∈L^∞(Rⁿ)×L^r(S^n-1)∩(0.0.15), r＞(?), fromH¹(Rⁿ) to L¹(Rⁿ) under the L¹-Dini condition and of the weak type (1,1) undercertain Dini condition. By interpolation, they obtained the L^p boundedness ofμ_Ωfor 1＜p＜2. For the operatorμ_Ω^ρ, from [95], we can get its L² boundedness,when 0＜ρ＜n,Ω∈L^∞(Rⁿ)×L^r(S^n-1)∩(0.0.15), r＞(?). Ding and Li[41] also showed thatμ_Ω^ρ(0＜ρ≤n/2) is bounded on L²(Rⁿ). In 2007, Li [66]studied the L^p boundedness ofμ_Ω^ρ(0＜ρ＜n) for 1＜p≤2, by assuming thatΩ∈L^∞(Rⁿ)×L^∞(S^n-1) satisfies (0.0.15) and the following condition: In this chapter, we adopt an idea from [27], and obtain some mixed normestimates for the vector-valued operators. Applying this conclusion, we draw thatthe L^p boundedness for the operatorμ_Ω^ρ, ifΩis an odd function in the secondvariable. On the other hand, we extend and improve some results in [40]. Themain results are as follows:Theorem 0.0.10 Suppose that 0＜ρ＜n,Ω∈L^∞(Rⁿ)×L^r(S^n-1), r＞(?)and satisfies (0.0.15). IfΩ(x,y') is odd in the second variable y'. Then for1＜p≤max{(?),2}, then there exists a constant C＞0 such thatTheorem 0.0.11 Suppose that 0＜ρ＜n,Ω∈L^∞(Rⁿ)×L^r(S^n-1), r＞(?),satisfies (0.0.15) and the L¹-Dini condition. Then for 1＜p≤2, we havewhere the constant C is independent of f.