Estimates, Of Several Types Of Operators In Some Function Space

The boundedness of operators is a very important topic in harmonic analysis and PDE. So many problems are related to it, such as the convergence of the Fourier series, the posedness of the solution of some partial differential equations, and so on. This PH.D thesis focuses on the estimates of several operators on some function spaces. The paper contains five chapters.Chapter 1 gives the boundedness of the fractional integral operator Iαand the bilinear fractional integral operator Ba on the modulation space Mp,q(Rn). It's well-known that there are many operators are not bounded on the classical Lebesgue space Lp(Rn), so we take some other function spaces that are good substitutions for it. One of them is the modulation space MP,q(Rn) whose defini-tion will be given in the second section of this chapter. The space Mp,q(Rn) was attracted a lot of attention in recent years. The reader can refer to the articles [7,19,60,61,63,64,66], among numerous references on this space and its applica-tions. Particularly, from a recent work, we know that the unimodular Fourier multiplier Sβ(t)＝eit|Δ|β/2 arc bounded on Mp,q(Rn) for all 1≤p,q≤∞. (see also [5,9,10,11,12,45].) The operator Sβ(t)＝eit|Δ|β/2 are Fourier multipliers with symbol eit(|ξ| withβ> 0 and t∈R, and they are interesting in both harmonic analysis and partial differential equations. It is well known that S1(t), S2(t) and S3 (t) are closely related to the wave equation, the Schrodinger equation and the Airy equation respectively. Sβ(t) generally do not preserve any Lebesgue space Lp(Rn), except for p＝2. Thus, to study the boundedness of Sp(t), the modulation spaces is a good alternative class of the Lebesgue spaces. With this observation, one naturally expects that not only the operator Sβ(t), but also many other operators might have different boundedness property on the modula-tion spaces from those in the Lebesgue spaces. One important operator that will be studied in this chapter is the fractional integral operator Iα, which is defined by whereNext theorem is the basic estimates of Iαon the Lebesgue space Lp(Rn) and the Hardy space Hp(Rn). Here, we recall that Lp(Rn)= Hp(Rn) if 1< p<∞.Theorem A ([16]) Let Hp(Rn) denote the Hardy space,0< p<∞. Then for 0<α< n, the fractional integral Iαis bounded from Hp1(Rn) to Hp2(Rn) withIn a recent paper [60], Sugimoto and Tomita extended Theorem A to the modulation spaces, and they obtained the following theorem.Theorem C ([60]) Let 0<α< n and 1< p1,p2,q1,q2<∞.The fractional integral operator Iαis bounded from Mp1,q1(Rn) to Mp2,q2(Rn) if and only ifComparing Theorem A with Theorem C, we find that the fractional integral operator Iαhas a larger range on the indices p1,p2 of the boundedness on the modulation spaces than those on the Lebesgue spaces. Thus, it is interesting to extend Theorem C without the restriction 1< p1,p2,q1,q2<∞. Based on this motivation, we will study the boundedness of Iαfrom Mp1,q1(Rn) to Mp2,q2(Rn) by removing the restriction 1< p1,p2,q1,q2<∞. For 0< p≤1, we use Hp norm to define the Modulation Hardy spaceμp,qs(Rn), andμp,qs(Rn)＝Mp,qs(Rn) when p> 1, see section 2 of this chapter for specific definitions. The following two figures show the ranges of pair (1/p,1/r) for the boundedness of the fractional integral operator Iα. In Figure 1 and Figure 2, the ray starting at (α/n,0) represents all pairs (1/p,1/r) for which the operator Iαis bounded from Hp(Rn) to Hr(Rn). The shady area including its boundary represents all pairs (1/p,1/r) for which the operator Iαis bounded fromμp,q2(Rn) toμr,q1(Rn) with 1/q2≤1/q1＋α/n. Also, by using an equivalent definition of the modulation space, our method can simplify the proof of the sufficiency part in Theorem C given in [60]. Meanwhile, we extend the necessary part. Let's give our results.Theorem 1.1 Let 0< p≤1,0< q<∞,0<α<n. We have for all 0<np/n－αp≤r≤∞.Theorem 1.2 Let 0<α< n,1< p<∞,0< q1<∞,0< q2≤∞, and then Iαis bounded from Mp,q1(Rn) to M∞,q2(Rn). Conversely, if p> 1,0< q1<∞,0<q2≤∞, and Iαis bounded from Mp,q1(Rn) to M∞,q2(Rn), thenIn [27] and [33], the authors introduced the bilinear fractional integral op-erator Bαwhich is a generalization of Iα, whose definition is They obtained the estimates of Bαon the Lebesgue space Lp(Rn).Theorem B ([27] [33]) Assume that and f∈Lp1(Rn),g∈Lp2(Rn) with 1< p1, p2<∞. Then(1) if pi> 1, i= 1,2,(2) if Pi> 1. i= 1,2, and either p1 or p2 is 1, Analogue to Iα, we get the following estimates of Bαon Mp,q(Rn).Theorem 1.4 Let 0<α<n,1< p<n/n-α,0< q<∞, then In particular, for 1< p< n/n-α, p≤q, thenIn Chapter 2, we study the estimates of three classical summation operators and their maximal operators on the Triebel-Lizorkin space Fpα,q(Rn) with 0< p< 1. They are Poisson summation PN, Gauss summation GN and Bochner-Riesz summation BNδ, also their maximal operators are P*, G*,B*δrespectively. For N≥1,PN,GN, BNδare defined by And their maximal operators are We know that the above three operators and their maximal operators paly an important role in studying the spherical summation of the Fourier series. In order to obtain the convergence in a (quasi-)Banach space X, we usually need to show their boundedness on X. Moreover, if we want to show the almost everywhere convergence for f∈X, the boundedness of their maximal operators on X should be considered. In this paper, we consider the case X is the Triebel-Lizorkin space Fpa,q(Rn) (0< p< 1). In [43], Lu and Yang proved the boundedness of BNδon Fpa,q(Rn).Theorem D ([43]) Let a∈R,0< p< 1,0< q<∞, andδ> J-n+1/2 min{a,0}, where J=n/min｛p,q｝. Then BNδf is bounded on Fpa,q(Rn). Namely. where C is independent of N and f.Inspired by this theorem, we want to get the similar results for P* and G*. On the other hand, we know that PN and GN are bounded on the Hardy space Hp(Rn) while Hp(Rn)≈Fp0,2(Rn), so there is a natural question can we extend the boundedness to the Triebel-Lizorkin space Fpa,q(Rn)? Up on the two thoughts, we get the following result.Theorem 2.1 Let a∈R,0< p≤1,0≤q≤∞. Then PN, are bounded on Fpα,q(Rn). Namely, where C are independent of N and f.The second section of this chapter is to study the estimates of the maximal operators P*, G* and B*δr on Fpα,q(Rn). For B*δ, the existed result was proved by Stein, Taibleson and Weiss, which is a weak type boundedness on HP(Rn).Theorem E ([59]) Let 0< p< 1 andδ=n/p-n+1/2. Then where C is independent ofλ> 0 and f∈Hp(Rn). As the original idea of Theorem 2.1, we want to extend the result to Fpα,q(Rn). Luckily, by a detailed observation of the atomic decomposition of Fpα,q(Rn), we get our second result.Theorem 2.2 Letδ>n-1/2,α> n, n/α+δ+n+1/2＜p＜min{n/α, q}. Then where C is independent of f∈Fpα,q(Rn).Remark 2.3 In fact, the requisition ofδin Theorem 2.2 isδ> max{n/p－n+1/2-α,n-1/2} for a fixed p. It's easy to see thatδ=n/p-n+1/2 satisfies it. Althoughα> n may be so strong, we impose a weaker restriction onδthan Theorem E. For example, let n= 3, p＝1/100, q＝2, thenδ＝298 in Theorem E andδ> 298-αwith anyα> 3 in Theorem 2.2.Remark 2.4 If we change B*δto P* or G* in Theorem 2.2, the same con-clusion holds. This can be easily seen from the proof of Theorem 2.2.Although we have obtained the estimates of P* and G* in Remark 2.4, we are not satisfied with this results. Hence, we reconsider this case and get the strong estimates of P* and G* on Fpα,q(Rn). The tool will be used is a new atomic decomposition of Fpα,q(Rn) found by Han, Paluszynski and Weiss [29].Theorem 2.5 Let 0<p<1<q<∞,0<α< 1, then P* and G* are bounded from Fpα,q(Rn) to L∞(Rn).Chapter 3 gives the boundedness of the Hausdorff operator Hφon three function spaces, namely, the local Hardy space h1(R), the Triebel-Lizorkin space Fpα,q(R) (0< p≤1) and the modulation space Mp,q(R). Given a function cp defined on (0,∞), Hausdorff operator Hφis defined by Whenφ(ξ)＝α(1-ξ)α-1χ(0,1)(ξ), Hφis the so-called Cesaro operator Cαwith orderα.Historically, for the periodic case, Hardy [30] proved that ifΣn＝0∞ancosnx is the Fourier series of a function in Lp(0,π), then so isΣn=0∞(Tα)ncosnx for 1≤p≤∞, where(Ta)0=α0, (Tα)n=α1+α2+…+αn/n, n= 1,2,…, and the same is true for sine series. Kinukawa and Igari [34] showed that ifΣn＝1∞bnsinnx is a Fourier series, then the conjugate seriesΣn＝1∞(Tb)ncosnx is a Fourier se-ries. Siskakis [55] obtained the same type of theorem in the Hardy space on the unit disc, that is, the operator defined as C*f(z)=Σn＝0∞[(n+1)-1Σk＝0nak]zn, f=Σk＝0∞akzk,is bounded on H1(D). For the real line case, Goldberg [26] in-vestigated the properties of the operator Hφon the spaces LP(R) with 1≤p≤2. Georgakis [23] studied the Fourier analytic properties of Hφon the space of com-plex bounded regular Borel measures on R, and as a special case he showed that ifφ∈L1(R), then Hφis a bounded operator on L1(R).In recent decades, people were concerned with the boundedness of the Hausdorff (Cesaro) operators on the Hardy space HP(R) (0< p< 1). In order to state conveniently, we define two numbers as Obviously, if Aφ,p<∞, then Hφis bounded on LP(R). Moreover, Liflyand and Moricz got the boundedness of Hφon H1(R) as follows.Theorem J ([39]) If Aφ,1<∞, then Hφis bounded on H1(R).So we can conclude that Hφis bounded on both L1(R) and H1(R) when Aφ,1<∞. But we know that the local Hardy space h1(R)(see the first section of this chapter for it's definition), the Lebesgue space L1(R) and the Hardy spce H1(R) have the following embedding relations: From those facts, we may naturally ask whether Hφis bounded on h1(R) or not? Our first result answers this question.Theorem 3.1 If Aφ<∞, then Hφis bounded on h1(R).For 0< p< 1, Liflyand and Miyachi studied the boundedness of Hφon the Hardy space HP(R) and obtained the following theorem. Theorem K ([40]) Let 0< p< 1, M= [1/p—1/2]＋1.Ifφ∈CM, supp 0 is a compact subset of (0,∞), then Hφis a bounded operator on HP(R).As the original idea of Theorem 2.3, we want to extend the above theorem to Fpa,q(R). And the second result is obtained.Theorem 3.2 Let 0<p<1,0< q<∞,α∈R, J=1/min(p,q), [α]+ max(0, [α]), L is the smallest integer which is larger than max(J－2, [J－1－a]). Ifφ∈CL+2+[α]+, supp 0 is a compact subset of (0,∞), then there is a constant C depends on p, q, a andφ, such that for any f∈S(R), we have Furthermore, Hσcan be extended as a bounded operator on Fpα,q(R).We know that studying the boundedness of non-convolutional operators on the modulation spaces is a meaningful problem. And Hφis a non-convolutional operator. Hence, we discuss the estimates of Hφon the modulation spaces as the last part of this chapter. By the definition and properties of the modulation spaces in Chapter 1, we can easily get some estimates. Since the dilation trans-form formula on the modulation spaces is so complex, here we only write two theorems.Theorem 3.3 Let 1< p< q<∞, Then there is a constant C independent of 0 and f such thatTheorem 3.4 (i) Let 1< p< 2, if 0 satisfies then Hφis a bounded operator on Mp,p(R). (ⅱ) Let 2≤p≤∞, ifφsatisfies then Hφis a bounded operator on Mp,p(R)The purpose of Chapter 4 is to give the boundedness of the maximal function Mw(|f|p)1/p and M f on the weighted Orlicz-Morrey spaces, and also we obtain the necessity of the boundedness of Mw(|f|p)1/p.As We know that the Orlicz spaces is a good substitution for the Lebesgue spaces. It's role can be reflected in describing the boundedness of the Hardy-Littlewood maximal function on some function spaces near L1(Rn) [15,35,36]. In [46], Morrey first introduced the Morrey spaces to estimate the solution of PDE. Later, many people such as Chiarenza and Nakai studied the boundedness of the Hardy-Littlewood maximal function on the Morrey spaces [14,47,49]. Recently, Nakai introduced the Orlicz-Morrey sapces which can be seen as a unity of the Orlicz spaces and the Morrey spaces, and also he obtained the necessary and sufficient conditions for the boundedness of the Hardy-Littlewood maximal function on it as following.Theorem L ([48]) LetΦ,Ψ∈y andφ,ψ∈G.Then the following are equivalent:(ⅰ) There exists a constant A≥1, such that for all (ⅱ) The Hardy-Littlewood maximal operator M is bounded from LΦ,φ(Rn) to LΨ,ψ(Rn).In [8], Bloom and Kerman obtained the boundedness of M f on the weighted Orlicz spaces. On the other hand, Komori and Shirai discussed the boundedness of M f and Mωf on the weighted Morrey spaces in [38]. From those results, the purpose of this chapter is to get the boundedness of Mω(|f|p)1/p and M f on the weighted Orlicz-Morrey spaces while the weighted Orlicz-Morrey spaces can be considered as a unity of the weighted Orlicz spaces and the weighted Morrey spaces. At first, we give some necessary definitions. Let's define two sets of functions as G={φ:(0,∞)→(0,∞):φis almost decreasing,φ(r)r is almost increasing}, For a Young functionΦ,φ∈G, a weight functionωand a ball B in Rn, letDefinition 4.4 For a Young functionΦ,φ∈G and a weight functionω, the weighted Orlicz-Morrey spaces defined byIfω≡1, the above spaces are the Orlicz-Morrey spaces introduced by Nakai. For a Young functionΦand p∈[1,∞), letΦ(r):=Φ(rp), thenΦis a Young function. Moreover, ifΦ∈y,so doesΦ. Next, we give the maximal operators will be studied.Definition 4.9 For any p∈[1,∞), we define the weighted maximal operator by And M(f) denotes the classical Hardy-Littlewood maximal operator.For weight functions, we still use the symbol Ap (1≤p<≤∞) for the classical Muckenhoupt weights, whose definitions can be seen in the second section of this chapter. Meanwhile, we introduce the following set of weights. Obviously, A1(?)A*. Now, we give our main results.Theorem 4.1 LetΦΨ,y,φ,ψ∈G,W∈A∞,p∈[1,+∞). If there is a constant A> 0 such that for all we have Then Mwψρ)1/p s bounded from L(Φ,φ,w(Rn) to LΨ,ψ,w(Rn).Corollary 4.2 Under the conditions of Theorem 4.1, then the same bound-edness holds for Mw(|f|r)1/r (r∈[1,ρ) and M(f).Theorem 4.3 LetΦ∈Δ2,Ψ∈y andφ,ψ∈S,ρ∈[1,+∞). If Mw(|f|p)1/p is bounded from LΦ,φ,w(Rn) to LΨ,ψ,w(Rn) with w∈A* and w(Rn)=∞,then there is a constant A>0 such that (4.1) and (4.2) hold.Chapter 5 gives some properties of the zero modes and the zero resonances of the Dirac operators. Let's give the definition of the Dirac operator: whereα= (α1,α2,α3) is the triple of 4x4 Dirac matrices with the 2×2 zero matrix 0 and the triple of 2 x 2 Pauli matrices Q(x)=(qij(x)) is a 4x4 Hermitian matrix-valued function decaying at infinity. In fact, the Dirac operators generalize the classical Weyl-Dirac operators, see [4]. In [22], it was found that the existence of the zero modes (i.e.,eigenfunctions with the zero eigenvalue) of the Weyl-Dirac operators play a crucial role in the study of the stability of Coulomb systems with magnetic fields. Loss and Yau are the first to construct the zero modes of the Weyl-Dirac operators and their results were usefully applied in [22]. From the view of mathematics and physics, we know that the zero modes have a lot of applications [1,2,4]. For recent works, the reader can see [50,53]. Generally, we assume the following varnishing condition about the matrix Q.Condition 5.1 Each element qij(x)(i,j＝1,…,4) of Q(x) is a measurable function satisfyingDefinition 5.2 By a zero mode, we mean a function f∈Dom H=H1(R3) which satisfies By a zero resonance, we mean a function f∈L2,-S(R3)\L2(R3) for some s>0, which satisfies H f=0 in the distribution sense.In [54], Saito and Umeda gave the following results about the zero modes and the zero resonances of H.Theorem M ([54]) Suppose Condition 5.1 is satisfied with p>1. Let f be a zero mode. Then (ⅰ) the inequality holds for all x∈R3, where the constant C= Cρ,f depends on f and p. (ⅱ) the zero mode f is a continuous function on R3.Theorem N ([54]) Suppose Condition 5.1 is satisfied withρ>3/2 If f belongs to L2,-S(R3) for some s with 0< s< min{3/2,ρ—1} and satisfies Hf＝0 in the distribution sense, then f∈H1R3).The aim of this chapter is to extend the mentioned results by using a simple method, just to weaken the vanishing condition about Q. Here, we indicate that our results can not be obtained by the primary method in [54]. Theorem 5.3 Suppose Condition 5.1 is satisfied withρ∈(1/2,1]. Let f be a zero mode. Then (ⅰ) for anyρ'<ρ, there is a constant C＝Cρ,ρ'> 0 such that for all x∈R3, we have (ⅱ) the zero mode f is a continuous function on R3.Theorem 5.4 Suppose Condition 5.1 is satisfied withρ> 1. If f belongs to L2,-s(R3) for some s with 0< s≤min{3/2,ρ－1} and satisfies Hf=0 in the distribution sense, then f∈H1(R3).