Some Singular Integral Operator

The content of this paper is divided into four chapters.In the first chapter, we investigate theL^p estimate between non-tangential maximal function and area integral function of f on both the unit sphere S^n-1 and the product spheres S^n-1╳ S^m-1.In the second chapter, we study the singular integral operator JΩ,a and the Marcinkiewicz integral operatorμΩ,a. The kernels of the operators behave like |y|^-n-a(a > 0) near the origin, and contain an oscillating factor eⁱ|y|-β(β> 0) and a distributionΩon the unit sphere S^n-1. We prove that,ifΩis in the Hardy space H^r(s^n-1) with 0 < r = (n-1)/(n - 1 + r)(r > 0) , and also satisfies certain cancellation condition,then TΩ,a andμΩ,a extend bounded operator from the Sobolev space L_r^p to the Lebesgue space L^p for some p.In the third chapter, we considered the L^p boundedness of the commutator generated by a RBMO(μ) function and the singular integral whose kernel satisfies certain minimum regularity condition. Whereμis a Radon measure on R^d which only satisfiesμ(B(x, r))≤Crⁿ for any x∈R^d and r > 0 and some fixed positive constants C and n∈(0, d].In the fourth chapter, we introduce a new class of weights A_p(Rⁿ), which is contained in the classical A_p(Rⁿ). We will show that this weight class also satisfies some similar important properties with A_p such as the duality, the re- verse HSder's inequality and the splitting theorem ect. Then using these A_p. we shall prove some weighted inequalities and vector valued inequalities for certain singular integrals and maximal singular integrals with rough kernels.0.1 Equivalence properties of maximal function and area function on product spheresLet Bⁿ = {z∈R n: |z| < 1} be the unit ball in the Euclidean space Rⁿ, and let S^n-1 be its boundary, S^n-1 = {w¹âˆˆRⁿ : |w'| = 1}. As usual L^p(S^n-1 denotes he space of all measurable functions on S^n-1, with ||f||_p = (∫_sn-1|.f(w')|^pdw')1/p, where dw' denote the element of surface area on S^n-1. let z E Bⁿ, the function of X' is the Poisson kernel for the sphere S^n-1 (here and in the sequel, |z - x'| will denote the distance of the two points in Rⁿ).Suppose C^k(Sⁿ) is the space of k-times continuously differentiable functions on Sⁿ, Definition 0.1 Denote (?)(Sⁿ) =∩_k=0^∞C^k(Sⁿ), and (?)*(Sⁿ) is its dual space, the space of distributions.It is to easy to see that P_z(x')∈(?)(Sⁿ). If f is a distribution, u(z) =< f((?)), P_z((?)) > is a harmonic function in Bⁿ and is called the Poisson integral off.Suppose u is a harmonic function on Bⁿ, the non-tangential maximal func- tion and the area integral function of are defined by z∈Γa(x') whereΓ_a(x') is the convex hull of the point x'∈S^n-1 and the ball {w∈Rⁿ: |w| < sin a}, 0 < a <Ï€/2. In this chapter, we obtain Theorem 0.1 Let f f∈(?)*(S^n-1), u is the Poisson integral of f, 0 <α,β<Ï€/2, then for any 0 < p <∞, we have Aa(u)||L^p(S^(n-1)≤C₂||N_β(u)||Lp(S^n-1) (3) where C₁ and C₂ are positive numbers depending on n, pαandβ.Now, let we denote x = (x₁, x₂)×B^m = B, x' = (x'₁, x'₂)∈S^n-1×S^m-1 = S,â–½=â–½₁×▽₂ and P_y = P_y1P_y2, whereâ–½₁ andâ–½₂ denote the gradient operator separately on Bⁿ and B^m. As (?)*(S), we can define (?)*(S). For f∈(?)*(S), denote u(y) =< P_y((?)),f((?)) >, which is called the bi-Poisson integral of f. Thus u(y) is a biharmonic function on the product space B, that is to sayâ–³_u =â–³₂u = 0, whereâ–³₁ and Delta₂ denote the Laplace operator separately on Bⁿ and B^m.Suppose u is a biharmonic function on B, letα= (α₁,α₂),Γ_α(x') =Γ_a1(x'₁)×Γ_a2(x'₂), the non-tangential maximal and area integral function op- erators on product space are defined as follows N_β(u)(x')= sup |u(y)| (4) Then we haveTheorem 0.2 Suppose f∈(?)*(S), u is the bi-Poisson integral of f, letα,β∈(0,Ï€/2)², then for 0＜p＜∞, there holds ||A_a(u)||Lp(S)≤C|| N_β(u)||Lp(S) (6)where C is a positive number depending on n, m, p,αandβ.0.2 Rough hypersingular integral operators with an oscillating fac- torIf a distributionΩon S^n-1 satisfying＜Ω,Y_m＞=0 (7) for all spherical harmonic polynomials Y_m whose degrees m≤[(?)]((?)＞0), the greatest integer part of (?) = (n - 1)((?)^-1 - 1), we sayΩsatisfies the cancelation condition, where <, > denotes the paring on S^n-1.Definition 0.2[5] If f∈(?)*(S^n-1), u(z)=< f((?)), P_z((?)) >, z∈Bⁿ. Let P+f(x') = sup |u(rx')|, 0≤r＜1 then for any 0 < p <∞, the Hardy space H^P(S^n-1) is the set of {f∈(?)*(s^n-1): ||f||H^p＜∞}, whereDefinition 0.3[5] Letα＞0, k≥α, the Lipschitz space∧^a(S^n-1) is the set of all g∈L∞(S^n-1) with norm by [5], (H^q(S^n-1))* =∧^a(S^n-1), a = (n—1)(1/q-1).Fix a radial functionΦ∈C^∞(Rⁿ) satisfying supp(Φ) C {x∈Rⁿ : 1/2 < |x|≤2},0≤Φ(x)≤1, andΦ(x)＞c>0 if 3/5≤|x|≤5/3.LetΦ_j(x)=Φ(?) Define the functionΨ_k byΨ_k((?)) =Ψ_k((?)), SO that Definition 0.4 for 1 < p <∞an a∈R, the homogeneous Sobolev spaces L_a^p(Rⁿ) is the set of all distributions f satisfyingDenote, for t∈R and x∈Rⁿ, FΩ,t(f)(x) = 2^-t∫₀^2tb(s,2^t)<Ω,f_x,s> ds (8) where f-(x,t)(Y') = f(x-y) = f(x-ty') with t = |y| and b(s,t) be a bounded function on R⁺×R₀The Marcinkiewicz integral operatorμΩ,αand the singular integral operator TΩ,αare defined on Schwartz space S(Rⁿ) , respectively, byμΩ,α(f)(x) = (∫_R^{2-2ta|FΩ,t(f)(x)|2dt}1/2 (9) T_Ω,α(f)(x) =∫₂^-taFΩ,t(f) (x)dtBy the discussion in [7], we know that f∈S(Rⁿ) implies f_x,t(y')∈∧^a(S^n-1). Therefore, the above definitions are well-defined. Also, it is easy to check that if let b(s,2^t) = h(s)×(1/2,1)(s^2-t),where h(s) is an L∞function, then, up to a constant, TΩ,α(f)(x) is the same as the singular integral operator :In the caseα= 0, TΩ,αf(x) is the clasical singular integral operator which was studied by many authors.The reader can see [17, 18, 19, 12, 22, 23, 24, 25, 26, 29, 37, 39], among numerous references. In the caseα＞0, recently, Chen, Ding and Fan in [4] established the following theorem:Theorem A Let 1＜p＜∞, forα≥0,suppose thatΩ∈H^r(S^n-1) with r = (n - 1)/(n - 1 +α) and satisfies (1) for all {Y_m(y')} with degree m≤[α]. Then ||TΩ,αf||Lp(Rⁿ)≤C||Ω||H^r(s^n-1)||f||L_α^p(Rⁿ) (11) ||μΩ,αf||Lp(Rⁿ)≤C||Ω||H^r(s^n-1)||f||L_α^p(Rⁿ) (12) for all f∈S(Rⁿ), where L_α^p(Rⁿ) denotes the homogeneous Sobolev space, and C is a constant independent of f andΩ.People are also interested in the operator with an oscillating factor e^(it)-β in its kernel since it is related to the Bochner-Riesz operators(see [44]). Recently Chen, Fan and Le [11] consider the operator and obtained the Mlowing result:Theorem B LetΩ∈H^r(s^n-1) with r = (n-1)/(n- 1 +Î¥)(Υ＞0) and satisfies (1) for all {Y_m(y')} with degree m≤[Î¥], Then ||JΩ,αf||Lp(Rⁿ)≤C||Ω||H^r(S^n-1)||f||L_Î¥^p(Rⁿ) for allβ/(β+Î¥-α)＜p＜β/(α-Î¥), provided thatβ＞2(α-Î¥)≥0 and 0＜Υ≤α.Question: Can the range of p in Theorem B be enlarged? Is there an analogy of Theorem B on the Marcinkiewicz integral operator with the kernel which has an oscillating factor e^i|y|-β? So we study the hypersingular integral operator In this paper we will give a positive answer for the above question and the following theorems are the main results:Theorem 0.3 LetΩ∈H^r(S^n-1) with r = (n-1)/(n-1+Î¥)(Î¥> 0) and satisfy (1) for all {Y_m(y')} with degree m≤[Î¥]. Suppose b(s, t) is a bounded function on R+×R and satisfies∫₀²|(?)/((?)t)b(s,t)|ds≤C, for every t∈[1/2,2]. Then for all 2β/(2β+Î¥-α) < p < 2β/(α-Î¥), provided thatβ>α-Υ≥0 and 0＜Υ＜α. Theorem 0.4 LetΩand b(s, t) be given as in Theorem 1.1, then for allβ/(β+Î¥-α) < p <β/(α-Î¥), provided thatβ> 2(α-Î¥)≥0 and 0＜Υ≤α.Remark: In the proof of Theorem 0.4 and Theorem 0.5, we only use the case of k = 1 of Van de Corput's lemma, but in Theorem B the authors use the case of k = 2, so we obtain a larger range of p and simplify the proof. While we can't use the Fourier transform to obtain the L² estimate of the operatorμΩ,αas usual, here we use a method different from [11].0.3 Lp estimate for the commutators of singular integrals with non- doubling measuresLetμbe a Radon measure on R^d satisfying the growth condition for any x∈R^d and r > 0, where C and n are positive constants and 0 < n≤d, B(x, r) is the ball with centered at x and radius r. The doubling condition onμ,μ(B(x, 2r))≤Cμ(B(x, r))((?)x∈R^d), is an essential assumption in most of results of classical Calderon-Zygmund theory. However. recently it has been shown that a big part of the classical theory re- mains valid if the doubling assumption onμis substituted by the linear growth condition. More literatures and related topics can be found in [33, 34, 35, 36, 46, 47; 48, 49, 50, 51].Definition 0.5 Let K(x,y)∈L_loc¹(R^d×R^d\{(x,y) : x = y}) is called a Calderon-Zygmund kernel if |K(x,y)|≤C/|x - y|ⁿ (19) and there exists 0＜δ＜1 such that |K(x, y) - K(x', y)| + |K(y,x)-K(y, x')|≤C(|x-x'|^δ)/(|x- y|^n+δ) (20) for |x - x'|≤|x - y|/2.The Calderon-Zygmund operator(CZO) asociated to the kernel K(., .) and the measureμis defined (at least, formally) by Tf(x) =∫K(x,y)f(y)dμ(y). The above integral may be not convergent for many functions f because the kernel K may have a singularity along x = y. For this reason, one introduces the truncated operators T^∈,∈＞0: K(x, y)f(y)dμ(y), T∈f(x) =∫-(|x-y|)＞∈)K(x,y)f(y)dμ(y), and then one says that T is bounded on Lp(μ) if the operators T_∈are bounded on Lp(μ) uniformly on∈＞0.Given a cube Q (?) R^d, let N be the smallest integer≥0 such that 2^NQ is doubling. We denote this cube by Q.Definition 0.6 Let p＞1 be some fixed constant. We say that f∈L_loc¹(μ) is in RBMO(μ) if there exists some constant C₃ such that for any cube Q(centered at some point of supp(μ)) and for any two doubling cubes Q (?) R in R^d, |m_Qf - m_Rf|≤C₃K_Q,R, (22) where and N_Q,R is the first integer k such that l(2^kQ)≥l(R). The minimal constant C₃ is the RBMO(μ) norm of f.Given b∈RBMO(μ), we can define the kth order commutator of T induc- tively Hu, Meng and Yang [30] generalized the result of Tolsa, and showed that if T is bounded on L²(μ), then for any positive integer k, T_b,k is bounded on Lp(μ) for any p with 1＜p＜∞. The purpose of this paper is to improve the result of Hu, Meng and Yang. We will show that if the regularity condition (20) is replied by for any R > 0, y,y'∈R^d such that |y- y'| < R and some r, 1 < r <∞, where p > 1. Then the boundedness of T on L²(μ) also implies that T_b,k is bounded on Lp(μ) for any p with 1 < p <∞.The main result of this paper is stated as follows.Theorem 0.5 Suppose that k is a positive integer, K(x, y) is a function on R^d×R^d\{x = y} which satisfies (19) and (23), T is the operator defined as above. If T is bounded on L²(μ), then for b∈RBMO(μ), we have ||T_b,kf(x)||Lp(μ)≤C||b||_(RBMO)(μ)^k||f||Lp(μ). Remark Obviously, condition (23) is weaker than (20); When k = 1, Hu, Meng and Yang [31] proved the Lp(μ) estimate for Tb under the condition for any R＞O and y,y' E Rd such that |y-y'|＜R. We see that if we take r = 1 in (23), then (23) is weaker than (23'). Unfortunately, by the method of this paper, we can't obtain the result with r = 1.0.4 Weighted inequality of singular integralsForΩΕL1(Sn-1) and satisfies some continuality conditions, some weighted norm inequalities is well known of the singular integral operator that is, for w(x) E Ap(Rn), we have the following inequality see [2] or Chapter 5 of [42].There are lots of nice properties of Ap weights and the weighted norm in-equality of the singular integral is applied widely. But if the kernel is rough, for exampleΩE Lr(Sn-1), and if we still consider the Ap weights, then the weighted norm inequality obtained is not ideal. Kavid K. Watson in [52] showed that ifΩE Lr(Sn-1) and w(x) E Ap/r' (24) holds only when r'≤p＜∞, r'≤p＜∞. In view of this, we hope to find a new class of weights, such that the properties of Ap weights as well as the weighted norm inequalities for more singular integrals still hold. Usually, weights are defined by certain maximal functions, for example the classical weights in Ap(Rn) are non-negative measurable functions w(x), such thatwhere M is the Hardy-Littlewood maximal function. Watson defined a class of Ap(Mμ)(ref. [53]) weights by where {μj}j is a sequence of Boral measures. Especially, when considering the singular integrals, we take For such w(x) E Ap(MΩ), Watson proved that the corresponding singular integral operators satisfyas long asΩr E Lr(sn-1) and r＞1.Duoandikoetxea introduced a class of radial weights (?)p(Rn), see [20]. De-fine w E (?)p(Rn) if and only if w(x) = v1(|x|)v1/2-p(|x|), where vi E A1(R+) is decreasing function or v2i E A1(R+),i = 1,2. This definition is based on the decomposition theorem of Ap. Duoandikoetxea showed that ifΩE L1(Sn-1) and odd or ifΩE Lr(Sn-1), r＞1 and even, then the corresponding weighted norm inequality (24) holds.Inspired by [20], we introduce a new class of weights (?)p C Ap, but the weights Ap are contained. We will show that these weights have lots of similar properties and some weighted norm inequalities for certain singular integrals and maximal singular integrals have the same form as Ap.In section 2, we will show that the weights (?)p still satisfy the duality property, inverse HSlder inequality, but for the splitting theorem, we only have Proposition 0.6 If v1(x), v2(x) E (?)(Rn), then w = v1v1/2-p E Ap(Rn).In section 3, we will show the following weighted inequality Theorem 0.7 SupposeΩ(y') satisfing∫sn-1Ω(y')dσ(y') = O and (i)Ω(y') E L1(Sn-1) and odd; or(ii)Ω(y') E Lq(sn-1), q＞1 and even.Then for any w(x) E (?)p(Rn), there exists a constant C(w,p,Ω), such that considering the operatorwhere Ry' is the l-dimension operator, then we have the following theorem Theorem 0.8 SupposeΩ(y') E L1(Sn-1), and if for some r≥1, (?)w(t) E Ar(R1) there arethen for any 1＜p＜∞and w(x) E Ap(Rn), we haveKeywords: Area Integral Function, Non-tangential Maximal Function, Hardy Space, Oscillating Factor, Singular integrals, Macinkiewicz Integral operators, Sobolev Space, k-th Commutator, Calderon Zygmund operator, (?)p-weights, Inverse HStder inequality, Weighted inequality...