Types Of Operators In Triebel-lizorkin Spaces On Bounded

This PH. D thesis focuses on the boundedness of some important integral operators on the Triebel-Lizorkin spaces, homogenous and inhomogenous. The operators we mainly consider here are the rough singular integrals, the maximal singular integrals, the Marcinkiewicz integrals and g-functions with rough kernels. The L^p-boundedness of the above operators has been studied thoroughly. Our theorems here can be regarded as the extension of those known results. However, since the homogenous Triebel-Lizorkin spaces are much wider than the classical Lebesgue spaces, more additional techniques will be presented in the development of this thesis.Chapter 1 of this thesis consists of the background of the subject. A reviewof some known results regarding the boundedness on the Triebel-Lizorkin spaces is presented. Before this thesis, the Triebel-Lizorkin boundedness is alwaysobtained for linear operators, especially those convolutional operators such as singular integral, oscillatory integral and Riesz potential. However there is almost no Triebel-Lizorkin boundedness regarding the maximal operators, the Marcinkiewicz integrals or more generally, the square functions. In Chapter 1, We also introduce some fundamental theorems which are needed in the developmentof the following chapters.In Chapter 2, we mainly discuss two classes of rough singular integrals and the boundedness on the homogenous and inhomogenous Triebel-Lizorkin spaces are obtained. The singular integral we study are defined byDefinition 0.0.1 LetΩ∈L¹(∑) and have mean value zero on the unite sphere. Then for any f∈S(Rⁿ), we defineThe L^p boundedness of this T_Ωhas been studied by large number of authors. For a good review of this subject, one is refer to [37]. We only list two of these results, in order to be compared with our main results in Chapter 2.Theorem 0.0.2 LetΩ∈L¹(∑) and have mean value zero. Suppose for someα> 0 we haveThenholds for all (2 + 2α)/(1 + 2α) < p < 2 + 2α.Theorem 0.0.3 LetΩ∈H¹(∑) have mean value zero on the unit sphere. ThenT_Ωis bounded on L^p(Rⁿ), 1 < p <∞.The above two theorems are proved in [36] and [31] respectively. Below are our main theorems of Chapter 2.Theorem 0.0.4 LetΩ∈L¹(∑) and have mean value zero on the sphere. Suppose(0.1) holds for allα> 0. Then T_Ωis bounded on F_p^β,q,β∈R,1p^β,q,β>0,11(∑) have mean value zero on∑. Then T_Ωis bounded on F_p^β,q,β∈R⁺ and F_p^β,q,β∈R whth 1p^β,qand F_p^β,q,denote the homogenous and inhomogenous Triebel-Lizorkinspaces. Note that when 1 < p <∞, F_p^0,2-L^p. Thus Theorem 0.0.4 and Theorem0.0.5 improve the L^p boundedness of Theorem 0.0.2 and Theorem 0.0.3. In the mean time, they improve the following known results regarding the boundedness of singular integrals on the Triebel-Lizorkin spaces in the sense of less restrictions on the kernelΩIn 1990, Han, Frazier, Taibleson and Weiss proved for the inhomogenous singular integral thatis bounded on F_p^β,q when p, q,βare subject to some restrictions and K(x, y) has a strong continuity, see [38]. For the homogenous singular integral T_Ω, Chen, Fan and Ying proved that it is bounded on F_p^β,q,β∈R, 1 < p, q <∞, assuming onlyΩ∈L^r(∑), r > 1, see [11]. Jiang in her doctoral thesis [49], obtained the same boundedness under the weak restrictionΩ(x')∈L ln⁺ L(∑). In fact she further proved some estimates on weighted Triebel-Lizorkin spaces. In [11], the authors also defined a class of hypersingular integralsand proved the following theorem.Theorem 0.0.6 For 1 < p, q <∞, we set p = max{p,p/(p-1)}, q= max{q, q/(q-1)} and r =n-1/n-1+α,α> 0. Let N be an integer such that 4(N +1) >pαq. IfΩ∈H^r(∑) satisfies m,Ω> = 0 for all sphere polynomials with degree less that N, then we haveNote that Theorem 0.0.5 is the special case of Theorem 0.0.6 if we setα= 0. However this case is not contained the Theorem 0.0.6. The proofs of our Theorems differ substantially from the previous ones. In [11] and [49], their proofs depend on a decomposition of the singular integral with the idea originally comes from the famous work of Duoandikoetxea and Rubio de Francia published in Invent. Math. in 1986 (see [27]). Our proofs in Chapter 2 rely on the following vector valued inequality The required boundedness then follows by the definition of the Triebel-Lizorkin spaces using the Littlewood-Paley decomposition, and the fact that T_Ωcommutes with any convolutional operator. On the other hand, it is widely known that inequality (0.2) can be obtained directly from a weighted estimate of the following kind. That iswhere N is some maximal function defined byΩand is bounded on any L^r, r > 1.Our proof of Theorem 0.0.4 follows strictly the clue described in the previous paragraph. In fact this method works for any convolutional operators such as oscillatory integral and Riesz potential, as long as we can prove some weighted estimates as (0.3) for those operators. WhenΩ∈H¹(∑), we are not able to prove (0.3) yet. However in the proof of Theorem 0.0.3, the vector valued inequality (0.2) is directly obtained by applying the rotation method.Chapter 3 is devoted to the square function and the Marcinkiewicz integral. We shall prove their boundedness on the Triebel-Lizorkin spaces with possible less restrictions on their kernels. First let us recall the definition of the classical g-functionwhereφis an L-P function (see section 1.2 for its definition) andφ_t(x) = t^-nφ(x/t). As we know, g_φis a bounded operator on any L^p, p > 1. When n≥3, we may setφ(x)=Ω(x)|x|^1-nχ_{|x|<1}(x).Then (0.4) gives the definition of the Marcinkiewiczintegralμ_Ω. In order thatμ_Ωis well defined, we have to further letΩ∈L¹(∑) and have mean valued zero on the unite sphere. The L^p boundedness ofμ_Ωwas first studied by Stein in [60], and followingly in [5], [65], [63], [25], [1] and [66]. Now it is known that under three type of conditions,μ_Ωis bounded on L^p, p > 1. These three conditions areΩ∈H¹(∑),Ω∈L(ln⁺L)^r(∑) with 1/2≤r≤1 andΩsatisfies the condition of Theorem 0.0.4.Despite the numerous results on L^p boundedness, no boundedness on the Triebel-Lizorkin space is wittered forμ_Ω, or even for the classical g-function. This may simply be regarded as the result of the nonlinearity ofμ_Ωor g-function. However, to further explore this issue, we consider the Sobolev space L_α^p which is also a special case of the Triebel-Lizorkin space F_p^α,q (by taking the index q=2). Roughly speaking, the indexαcounts the times we take derivatives. Noting that taking derivatives commutes with convolution, one may prove the boundedness on Sobolev spaces without much difficulty if a prior L^p estimate is known. This in fact has already been indicated in the proofs of Chapter 2. When considering the non-convolutional operators, we usually regard them as some vector valued convolutional operators. For example, the vector space corresponds to g-function is H = L²(R⁺,dt/t) while for the maximal operator it is H = L^∞(R⁺,dt). To obtain the Sobolev boundedness of these operators, we must find out a suitable way so that the derivative can "pass" the vector norm‖·‖_H.In 1997, Kinnunen proved the following theorem for the Hardy-Littlewood maximal function (see [53]).Theorem 0.0.7 Suppose f∈L₁^p(Rⁿ), 1 < p <∞. Then Mf∈L₁^p(Rⁿ) and further more,holds for almost all x∈Rⁿ.The above theorem enables us to take the first derivative on the Hardy-Littlewood maximal operator. The boundedness of M on L₁^p is then an easy consequence of this theorem. Later, Korry proved that the Hardy-Littlewood maximal operator is also bounded on L_α^p with allα∈[0,1], see [51] where the theorem is stated for more general nonlinear operators including the classical g-function. However, his method applies only for those with very smooth kernels. Our first task in Chapter3 is to improve Korry's theorem so the we can deal with the Triebel-Lizorkin spaces. Then by incorporating a rotation method applied for the Marcinkiewicz integral, we get the Triebel-Lizorkin boundedness forμ_ΩwhenΩ∈H¹(∑). In fact, the following more general theorem is proved in Chapter 3. Theorem 0.0.8 LetΩ(x')∈H¹(∑) and have integral zero on E. Suppose there are someε,γ> 0 such that |h(s)|≤Cs^-n+ε/(1+s)^2ε andThen the g-function g_φdefined byφ(x) = h(|x|)Ω(x) is bounded on F_p^β,q and F_p^β,q where 0 <β< 1 and 1 < p, q <∞.It is not hard to see that the g-function considered in Theorem 0.0.8 contains the caseμ_Ω. If a further continuity requirement is imposed on the radial function (Ωbeing still in H¹(∑)) of Theorem 0.0.8, we can prove a weighted estimate as well as a vector valued inequality for g_φ. This is put in the last paragraph of Chapter 3 as an application of our method. At this point, let us note that we are not able to prove the Triebel-Lizorkin boundedness via the vector valued inequality for a nonlinear operator.In the last chapter, we study the boundedness of some maximal operators on the Triebel-Lizorkin spaces. The proofs there are similar in nature to those in Chapter 3. The fact that maximal operator is an L^∞(R⁺,dt) valued convolution is used in a crucial way. The operators we consider includeand the maximal singular integralWith the only assumption thatΩ∈L¹(∑), M_Ωis bounded on F_p^β,qand F_p^β,q when 0 <β< 1 and 1 < p, q <∞. By the rotation method, we are restricted to deal with a one dimensional operator on which we invoke the method of Chapter 3. For the maximal singular integral, we letΩ∈H^∑ and have mean value zero on the unit sphere. Then the same boundedness is reached. Our proof there follows the clue of obtaining the L^p boundedness for that very operator. The difficulty then passes to the estimates of some one dimensional operators. For the sake of completeness, we also introduce some other operators and their boundedness on the Triebel-Lizorkin spaces in our last chapter. These operatorsinclude oscillatory integral, Riesz potential and some commutators generatedby BMO functions. Many of those works appear in Jiang's doctoral thesis [49]. A brief summary of the thesis is put in the last section where we also list some unsolved questions.