Boundedness Of Singular Integral Submanifold

In chapter 1, we study the mapping properties of the singular and maximal Radon transforms with rough kernels. The research of Radon transforms originated in Fabes' proof of the L²-boundedness of Hilbert transform along the parabola, which is defined byThe initial L^p results were then obtained by Nagel, Riviere, and Wainger (see [2],[3]). In the process of their proof, they found that the curvature condition which the curve satisfied was an important tool. Notice that in formula (0.14), the curveγ(t) is independent of the variable x. The operator H_γis in fact a convolution-type operator, with the property of translation invariance. Hence, some classic methods in harmonic analysis, such as Fourier transform and Plancherel formula, are still applicable for the proof of the boundedness of such operator. When the curveγ(t) changes with variable x or is replaced by "variable" surface or submanifold, then the operator defined by formula (0.14) is more complex. To prove its L^p-boundedness, some new tools and techniques are needed. In [4], Nagel, Stein and Wainger introduced the almost orthogonality method, which is effectively in proving the L²-boundedness of Radon transform. And this method has been used till now.In 1986, in order to solve Neumann problem on the pseudo-convex domain, Phong and Stein investigated the L^p-boundedness of the singular Radon transform along sub-manifold with nowhere vanishing rotational curvature. In their paper, the sub-manifold r(x, t) is defined bywhereΦ(t,x,y) is a real smooth function defined on Rⁿ⁺¹Ã—Rⁿ withΦ(t,x,x) = 0 for all (t,x)∈Rⁿ⁺¹. The nowhere vanishing rotational curvature is equivalent to the nonvanishing of determinant of the matrix Phong and Stein proved that if the kernel K(t, x, z) is a smooth function with fixed compact support and satisfies the conditionsThen the Radon transform defined byis bounded on L^p for 1＜p＜∞.Phong and Stein's results show that the mapping properties of the singular Radon transform are not only dependent on the curvature conditions of the sub-manifold, but also related to the regularity of the kernel. Recently, there are many researches in this area. We refer the reader to [6], [7], [8], [9], [10], [11], [5], [12], [13],[14], [15].In [6], Christ, Nagel, Stein and Wainger improved the results of Phong and Stein. They studied the mapping properties of the singular Radon transform defined byHere r_t(x) = r(x,t) is a smooth function, defined in a neighborhood of the point (x₀,0) in Rⁿ×R^k, taking values in Rⁿ with r(x, 0) = x. The map x(?)r_t(x) is a family oflocal diffeomorphisms of Rⁿ, depending smoothly on the parameter t. The differential(?) has rank k when t=0. It was proved in [6] that there exists a unique collection ofvector fields X_αdefined in some neighborhood of x₀ with (α₁…α_k)≠0, such thatThe curvature condition that the submanifold r(x, t) satisfies at a point x₀∈Rⁿ could be stated as follows:curvature condition r satisfies curvature condition(C) at x₀, if there exists m＞0 such thatwhere R(x,t)=O(|t|^m+1), and the vector fields {X_α:|α|≤m} together with all their iterated commutators of degree≤m span the tangent space to Rⁿ at x₀. In formula (0.15),ψis a suitable smooth cut-off function supported near x₀ and a is a sufficiently small positive constant. K(t)∈C¹(R^k\{0}) is homogeneous of degree -k and satisfiesMoreover, the author of [6] also investigated the iL^p-boundedness of the maximal Radon transform defined byIn [6], the authors have made two promotions in their conclusions. On one hand, the curvature conditions which the sub-manifold satisfies are weaker than that in [5]. On the other hand, the regularity of the kernel K(t) is weaker than the smooth condition that the kernel satisfies in [5].Inspired by Christ, Nagel, Stein and Wainger, in section 1.2, we prove the following resultTheorem 1.1 Suppose that r(x, t) satisfies the curvature condition (C) at x₀ and K₀ is in the Orlics space Llog L(A₀). Then the operator T defined by (0.15) is bounded on L²(Rⁿ).Here the set A₀ is defined by A₀={t∈R^k:(?)≤|t|≤α}. The function K₀(t) is the restriction of K(t) to the annuals A₀.In section 1.3, under weaker assumptions on the regularity of the kernel, we prove the L^p-boundedness of the singular Radon transform . Our result is as followsTheorem 1.2 Suppose that r(x, t) satisfies the curvature condition (C) at x₀ and K₀∈L^q(A₀) for some 1＜q＜∞. Then the operator T defined by (0.15) is bounded from L^p(Rⁿ) to itself for every 1＜p＜∞.In formula (0.17), if we introduce a rough kernel and consider the maximal Radon transform defined bywhereΩ(t) defined on R^k is homogeneous of degree 0. Then the operator M has the following mapping property Theorem 1.3 Suppose that r(x, t) satisfies the curvature condition (C) at x₀ andΩ(t)∈L^q(S^k-1)for some 1＜q≤∞. Then the operator M defined by (0.17) is bounded from L^p(Rⁿ) to itself for every 1＜p≤∞.For a compact set D, we have the following embeddingsSo in chapter 1, we improve the main results in [6]. We also give a partial solution to the question which is raised by the well-known analysts D.Muller in his characteristics comment on the paper [6]. Inspired by the arguments presented in [6], in our proof, we reduce the study of the singular Radon transform T defined by (0.15) to one on nilpotent Lie group by a lifting technique, where proving L^p-boundedness is less complicated. Then using the Littlewood-Paley decomposition and almost orthogonality methods, we obtain our main results.Time-frequency analysis is an important branch of Harmonic analysis. It is important to find a suitable way to characterize the time-frequency concentration of a signal or distribution. The Fourier transform provides only non-localized frequency information. However, the short-time Fourier transform or windowed Fourier transform is an important joint of time-frequency representation. It provides an effectively way to measure the time-frequency concentration of a distribution. Using the short-time Fourier trans-form, Feichtinger introduced a class of Banach spaces called the modulation spaces, denoted by M^p,q,p,q∈[1,∞].Let g be a non-zero Schwartz function and 0＜p,q≤∞and s∈R, the modulation space M_s^p,q(Rⁿ) is defined as the closure of the Schwartz class with respect to the normwhere V_gf(x,w) is the so-called short time Fourier transform (STFT), which is defined byi.e. the Fourier transform F applied to (?).Recently, the above definition has been generalized by Kobayashi [18] to the case0＜p,q≤∞. In his definition, the function g is restricted to the spaceΦ_α(Rⁿ). For α＞0, we defineΦ_α(Rⁿ) to be the spaces of all g∈S(Rⁿ) satisfyingWe may choose a sufficiently small a, such that the function spaceΦ_α(Rⁿ) is not empty. It is not difficult to see that when 1≤p,q≤∞, these two definitions are equivalent, while 0＜p,q＜1, modulation spaces are just quasi-norm spaces. For a more general definition, involving different kinds of weight functions, both in the time and the frequency variables we refer readers to section 2.2.Wiener amalgam space is also a kind of important space in time-frequency analysis. It is well known that the Fourier image of the modulation spaces are Wiener amalgam spaces. In addition, there exist some embedding results between them. For more details of this aspect, we refer the readers to section 2.3.During the last ten years, modulation spaces have become useful function spaces for both time-frequency and phase-space analysis. They have also been employed to study boundedness properties of pseudo-differential operators (see [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32]), multilinear pseudo-differential operators (see [33], [34], [35], [36]), Fourier multipliers (see [37], [38], [39], [40]), Fourier integral operators (see [41], [42])and well-posedness of solutions to PDE's(see [43], [44], [45], [46], [47]). Based on some oscillating integral estimates, in section 2.4, 2.5 and 2.6, we investigate the boundedness of the singular convolution operator and hypersingular integral operator along submanifolds on modulation spaces. Using Fourier transform, these operators can be viewed as Fourier multiplier operators. From our results, we can see that modulation spaces are good substitutions for Lebesgue spaces.LetDefine the convolution operator byIn 1970s, Fefferman, Stein and Wainger studied the L^p-boundedness of this operator (see [48],[49]). Their conclusions highlight that the L^p-boundedness of T is closely related with the range of the parameter a andα. According to the different values of a,α, we recall some results of it on Lebesgue spaces.In the case n = 1, a＞0 and a≠1,1-(?)＜α＜1, Sampson, Naparstek and Drobot obtained the following result. Set p₀=a/(a-1+α)and p₀' = p₀/(p₀-1), then T is bounded on L^p(R) if p₀＜p＜p₀' (see [50]). In [51], Persj(?)lin extended the above results to Rⁿ. He proved that if n(1-(?))≤α＜n and p₀ = na/(na - n +α), then the operator T is bounded on L^p(Rⁿ) if and only if p₀≤p≤p₀'. Using the fact L^p(Rⁿ)=H^p(Rⁿ) for 1＜p＜∞, Persj(?)lin reduced the proof of L^p-boundedness of the operator T to its H^p-boundedness. In section 2.4, we prove the following resultTheorem 2.1 Suppose a＞0 and a≠1,α≤n, then the operator T defined by (0.20) is bounded on M_s^p,q(Rⁿ) for 1≤p≤∞,0＜q≤∞and s∈R.Notice that in theorem 2.1, the parameter a only needs to satisfy the conditionα≤n, which is weaker than that in [51]. Moreover, the rang of the exponent p is larger than that in Persj(?)lin's result.In (0.19), if we takeα=-βand replaceαby n+α, then we get the following functionWhereηis a smooth, compactly supported, radial function equal to 1 in the unit ball. Ifα＜0, the kernel K_α^*(x) is integrable, hence the operator T^* is bounded on L¹. Whenα＞0, in [52], Wainger proved that T^* is bounded on L² if and only ifα≤(?). Interpolating this result with the L¹ result above, Wainger obtained that the operator is bounded on L^p forThe question of what happened at the endpoints (?)=(?) was settled later. The counterexample showing that this result is sharp is due to Wainger.Inspired by Wainger, in section 2.4, we investigate the boundedness of the convolution operator T defined by (0.20) on modulation spaces. Its kernel function is defined by Our result is as followsTheorem 2.2 Assumeα≤(?), then the operator T is bounded on modulation space M_s^p,q(Rⁿ) for 1≤p≤∞,0＜q≤∞and s∈R.It is well known that Hilbert transform along curveis bounded on Lebesgue spaces, if the curveγ(t) satisfies the appropriate curvature conditions. Such kind of operator has been studied extensively by many authors (see [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65] ,[66]).If we enhance the singularity at the origin and consider the operatoralong homogeneous curveγ(t)=|t|^k orγ(t)=|t|^k sgnt. Then it is not difficult to see that such operator is not bounded on L², since its Fourier multiplier is not uniformly bounded (see [54]). To counterbalance this worsened singularity, we introduce an additional oscillation e^{-2Ï€i|t|-β} and study the operatorthen the operator T_α,β is bounded on Lebesgue spaces .The operator T_α,β was initially studied by Zielinski in his Ph.D. Thesis (see [63]). He showed that ifγ(t)= t², then T_α,β is bounded on L²(R²) if and only ifβ≥3α. This result was later improved by Chandrana in [54]. He considered the general homogenous curvesγ(t)=|t|^k or |t|^k sgnt for k≥2 and proved that ifβ＞3α＞0, then the operator T_α,β is bounded on L^p(R²) forIn addition, he also obtained that the sufficient and necessary condition for the L²-boundedness of this operator isβ≥3α.Recently, the author of [57] extended the results of [63] and [54] to high dimensions and investigated the L^p-boundedness of the operator whereθ=(θ₁,θ₂,…,θ_n) andThey proved that T_n,α,β is bounded on L²(Rⁿ) if and only ifβ≥(n+1)α. And ifβ＞(n+1)α, then T_n,α,β, is bounded on L^p(Rⁿ) for(?)＜p＜(?).Inspired by Chandrana and the authors of [57], in section 2.5, we prove the following two results:Theorem 2.3 Letγ(t)= |t|^k orγ(t) = sgnt|t|^k, k≥2 and T_α,β be defined by (0.23),then(â…°) Ifβ≥3α, then the operator T_α,β is bounded on M_s^p,q(R²) for 1≤p≤∞, 0＜q≤∞and s∈R.(â…±) If T_α,β is bounded on M^2,2(R²), thenβ≥3α.Theorem 2.4 Suppose p₁,p₂…p_n are distinct positive numbers and the operator T_n,α,βis defined as (0.24), then(â…°) Ifβ≥(n+1)α, then T_n,α,β is bounded on M_s^p,q(Rⁿ) for 1≤p≤∞,0＜q≤∞and s∈R(â…±) If T_n,α,β is bounded on M^2,2(Rⁿ), thenβ≥(n+1)α.Comparing theorem 2.3, 2.4 with the results in [54] and [57], we can see that therange of the exponent p for modulation spaces is larger than that for space L^p(Rⁿ).Furthermore, the parameterβfor L^p-boundedness can not take the value (n + l)α.However, we can take it in our result. In particular, for n = 2, we can takeβ= 3α.On the other hand, the study of the hypersingular integral along surfaces is also anattracting subject in Harmonic analysis. In 2002, Hung Viet Le considered the following operatorHereΩis homogeneous of degree zero and has mean value zero over the sphere S^n-2. In his paper, Hung Viet Le proved that ifβ＞2α＞0 andΩ∈L^q(S^n-2) for some (1＜q＜∞),γ(y) satisfying some smooth conditions, then T is bounded on L^p(Rⁿ) for (?)＜p＜(?)(see[66]). Inspired by Hung Viet Le, in section 2.6, we prove the boundedness of the operator defined by (0.25) on modulation spaces. Our rough kernelΩis in space L¹(S^n-2), which contains the space L^q(S^n-2).Theorem 2.5 Assume that h(y) andγ(y) defined on R^n-1(n≥3) is real-valued,radial and differential a.e. on [0,∞). h(y) satisfies the conditions that h is continuous,bounded, and that either h is monotone or h'∈L¹(R).γ(r) satisfies the followingconditionsa) |γ'(r)| is increasing on suppγ'∩[0,∞) andeitherorc)γ∈L^∞(R) andγ(r) is monotone on [0,∞).Ωdefined on R^n-1 satisfies the following conditions:d)Ωis homogeneous of degree zero,e)Ωhas mean value zero over the sphere S^n-2, and Ifβ≥2α＞0, then T defined by (0.25) is bounded on M_s^p,q(Rⁿ) for 1≤p≤∞,0＜q≤∞and s∈R.Note that in Hung's results the conditions whichγ(y) satisfy are stronger than the homogeneous hypersurfaceγ(y) =|y|^k for y∈R^n-1. So in section 2.6, we also study the mapping properties of the operator defined by (0.25) along homogeneous hypersurface. In this case, the rough kernelΩis in L¹(S^n-2), too. Our second result in section 2.6 is as followsTheorem 2.6 Let the function h(y) defined on R^n-1 be real-valued, bounded, measurable, radial and differential a.e. on [0,∞). The function ft which defined on R^n-1 satisfies the following conditions:a)Ωis homogeneous of degree zero,b)Ωhas mean valued zero over the sphere S^n-2, andSuppose thatγ(y)=|y|^k and k is a positive real number, then the following results is true:If k≥2 andβ＞3α＞0, then T defined by (0.25) is also bounded on M_s^p,q(Rⁿ) for1≤p≤∞, 0＜q≤∞and s∈R.In the last chapter, we concentrate on some embedding results between modulation spaces M_s^p,q(Rⁿ) and Besov spaces associated with Schr(?)dinger operators H. To begin with, we introduce the smooth dyadic system.For j=0,1,…,the function sequenceφ_j∈C₀^∞(R) is a smooth dyadic system if it satisfies the following conditions:Using the smooth dyadic system, Triebel defined the inhomogeneous Besov space B_s^p,q(Rⁿ). Let0＜p≤∞,0＜q≤∞,and s∈R. If f∈B_s^p,q(Rⁿ), thenIt is well known that there exist some embedding results between Besov space B_s^p,q(Rⁿ)and Lebesgue spaces. These embedding results play important role in PDE (see[43],[68], [69], [30], [70]).In recent years, many scholars try to use Schr(?)dinger operator to characterize some common function spaces, such as Sobolev space, Triebel-Lizorkin space and Hardy space, etc (see [71], [72], [73], [74], [75], [76]). It is natural to ask a question if we replace the differential operator D by Schrodinger operator H, whether (0.26) is still well defined or not. In [77], the authors gave a solution to this question. They showed that for l = 0,1 and every N＞0, if there exites a constant C_N＞0, such that for allThen the operator H could be applied to characterize the Besov space with full range of parameters 0＜p,q≤∞. The definition of inhomogeneous Besov space associate with the operator H is as follows Definition 3.1 Let s∈R,0＜p≤∞and 0＜q≤∞, then the inhomogeneous Besov space associate with Schr(?)dinger operator is defined as the completion of the Schartz class S(Rⁿ) with the quasi-normHowever, if the operator H=-Δ+V(x) satisfies the following conditions: (â…°) V(x)≥0;(â…±) H satisfies the upper Gaussian bound for the heat kernel and its derivative. Then the kernel functionφ_j(H)(x,y) of the operatorφ_j(H) satisfies the formula (0.27). Hence, the inhomogeneous Besov space B_s^p,q(H) associate with such Schrodinger operator H is well defined. And there exist the following embedding results between modulation spaces M_s^p,q(Rⁿ) and them.Theorem 3.1 Let H=-Δ+V(x) be Schr(?)dinger operator and satisfy the following conditions: (â…°) V(x)≥0; (â…±) Forα=0,1, the heat kernel e^-tH(x,y) satisfies the following conditionsIf real number s₁,s₂,s₃,s₄ satisfy s₁ = (?)Then we have the following results...