| Harmonic analysis, which has profound history background and rich theory, is one of the most important branches in mathematics, and has been extensively applied in almost all field of mathematics. The singular integral which has been developed more than half century has a high position in harmonic analysis.This thesis focuses on the boundedness of Cauchy integral operator and Poisson integral operator on some functional spaces. It is divided into three chapters:In chapter 1, we study the boundedness of Cauchy integral operator on weighted Hardy space; In chapter 2, we study the boundedness of Cauchy integral operator on the predual of a Morrey space; In chapter 3, we get that the Poisson integral operator on sphere is bounded from Lebesgue space to Lorentz space. In the following, we will state the main content of every chapter.Chapter 1 Let Rn be the n-dimensional Euclidean space. The Cauchy integral operator is defined by where A (x) is a real valued function. This operator is very important in real and complex analysis, and has attracted many mathematicians to investigate it, see, for example,We all known that Lp(Rn)(p≤1) no longer has the same good nature as 1< p<∞. The Hardy spaces Hp(Rn) is ideal substitutes of them when p≤1. For example, when p≤1, it is well known that Riesz transforms are not bounded on Lp(Rn), however, they are bounded on Hardy spaces Hp(Rn). The theory of Hardy spaces plays an important role in theory of boundedness about various operators and partial differential equations, see, for examples, But, unfortunately, Hp(Rn) also has lots of defects. For instance, if f∈Hp(Rn),η∈C0∞(Rn), fηis not necessary in Hp(Rn)(see); Pseudo-differential operators are not bounded on Hp(Rn)(see ). The reason why these properties fail can be seen from the fact that∫f= 0 provided that f∈Hp(Rn). In order to overcome the shortcomings of Hp(Rn), Goldberg, in [32], introduced the local version of Hardy space, hp(Rn). Since then, hp(Rn) has been studied extensively, it also play an important role in the theory of boundedness about various operators and partial differential equations, see, for examples, [18,32,33,44,57]. Consequently, the weighted versions of Hp(Rn) and hp(Rn) have also been studied extensively, more systematic research in this regard can be founded in [58].Next, we give the definition of Calderon-Zygmund operators, which can be found in [31,56]. Since we are interested in the Cauchy integral operator, we will use the following definitions(see [42]).Definition 0.1.1 Let 0<δ≤1. A locally integrable function K (x,y) de-fined on{(x,y)∈Rn x Rn:x≠y} is called a Calderon-Zygmund kernel if it satisfies the following conditions for all 2|y - z|<|x - z|. We always denote it by K∈SK(δ).Definition 0.1.2 We say an operator T to be aδ-Calderon-Zygmund opera-tor associated with a Calderon-Zygmund kernel K (x, y) if for every f E L2 (Rn), exists almost everywhere on Rn and T is bounded on L2(Rn), i.e. C||f||L2. The transpose of T is denoted by For a bounded function f, we define Note that if∈L2 (Rn)∩L∞(Rn), then tTf (x)=tTf (x)+Cf a.e. where Cf is a constant.Definition 0.1.3 For 0<α≤1, the Lipschitz space Aα(Rn) and the local Lipschitz space∧locα(Rn) are the set of all functions f satisfying the following conditions respectivelyIt is easy to see that∧1 (Rn)=∧loc1 (Rn) and∧α(Rn) (?)∧locα(Rn) (0<α< 1), where the inclusion is proper. A simple example is f(x)= x, for which we can check that it belongs to∧locα(R1) but not to∧α(R1) (0<α< 1). Further-more, we know that the dual space of Hp (Rn) is∧n(1/p-1) (Rn), i.e. (Hp (Rn))*=∧n(1/p-1) (Rn), where n/n+1< p< 1(see).Definition 0.1.4 Letβ> 0. A bounded function b is said to beβ-accretive if Reb (x)≥βfor almost all x.In 1977, Calderon proved that CA is bounded on L2(R1) if|| A'||L∞is small. In 1982, Coifman, McIntosh and Meyer showed that the restriction of||A'||L∞being small is not necessary. That is:Theorem 0.1.1 If A'∈L∞(R1), then CA∈CZO(1).If A'∈L∞(R1) and w∈Ap(1< p<∞), it is easy to check that CA is bounded on Lp (R1) (1< p<∞) and weak type bounded on L1 (R1) on base of Theorem 0.1.1 and the theory of Calderon-Zygmund operators. Furthermore, according to the weighted theory of Calderon-Zygmund operators, if A'∈L∞(R1) and w∈Ap(1< p<∞), it is also easy to check that CA is bounded on Lwp(R1) and weak type bounded on Lw1(R1). More details can be founded in [24,31].The above discussion about CA focus on the LP(R1)(1≤p<∞). But few results are known on the Hardy space Hp (R1). Recently, in [42], Komori showed that CA is bounded from Hp (R1) to hP (R1)(the local Hardy space):Theorem 0.1.2 Let 0<∞< 1 and1-1+α<≤p≤1. If A'∈L∞(R1)∩∧α(R1), then CA is bounded from Hp (R1) to hp (R1).In order to prove the Theorem 0.1.2, the author considered a variant of "Tb theorem":Theorem 0.1.3 Let 0<α< 1,n/n+δ< p≤1 andn/n+α≤p, T beαδ-Calderon-Zygmund operator. If there exists aβ-accretive function b such that b,tTb∈∧α(Rn), then T is a bounded operator from Hp (Rn) to hp (Rn) andIn [42], the author obtained the hp(Rn)-estimate (0< p< 1) by(Hp (Rn))*=∧n(1/p-1) (Rn) and the h1 (Rn)-estimate by interpolation. In this paper, we apply different method to prove our theorems. The details can be seen in section 5 of chapter 1.A natural question is whether CA is bounded from Hwp (R1) to hwp (R1)? We will give the affirmative answer in section 3 of chapter 1. To prove the theorem, we introduce a generalized atom and molecules on hwp (R1) and consider a variant of weighted "Tb theorem". Our main results are as the following:Theorem 0.1.4 Let 0<α≤1≤q,nq/n+δ< p≤1 and nq/n+α≤p< q. As-sume that w∈Aq and T is aδ-Calderon-Zygmund operator. If there exists aβ-accretive function b such that b, tTb∈∧locα(Rn), then T is a bounded operator from Hwp(Rn) to hwp(Rn) andCompared with Theorem 0.1.3,we improve the scope ofαto 1 and weaken the restriction of tTb.Note that A1 (?) Aq(1 0, where C is the set of complex numbers. The most important example is given byφ(t)= tλ-1/q, whenλ= 0, Lq,φ(Rn) becomes Lq(Rn). Forλ= 1, the space Lq,φ(Rn) coincides with one of the equivalent versions of the space BMO(Rn) and for 1<λ< 1+q, the space Lq,φ(Rn) is known to coincide with Liptchitz space Aλ-1/q(Rn). Finally, for 0<λ< 1, the space Lq,φ(Rn) is the usual Morrey space which was introduced by Morrey in [48]. Since then, the type of Morrey space has been studied widely and it play an important role in partial differential equations and theory of sigular integrals, see, for example, [1,3,6,11,23,37,53,60].In 1986, Zorko[66] firstly proved that the predual of Lp',φ(Rn) is Hp,φ(R1), which means that the dual space of Hp,φ(R1) is Lp',φ(Rn)(the details can be founded in section 2 of chapter 2). And since then, there are many research papers about the new space, see, for instance, [2,45,64]. In 2003, Komori [40] proved that the Calderon commutator TA1 is bounded from Hp,φ(R1) to Hp,φ(R1), where hp,φ(R1) is the local version of Hp,φ(R1) and similar to local Hardy space. That is: Theorem 0.2.1 Let 0<α< 1< p≤1/1-α. If A'∈L∞(R1)∩∧α(R1), then TA1 is bounded from Hp,φ(R1) to hp,φ(R1).The author proved this theorem as a corollary of the following theorem:Theorem 0.2.2 Let 0<α<1 1) is also bounded from Hp,φ(R1) to hp,φ(R1).We have known that CA has close link with TA1(see ). A natural question is whether CA is bounded from Hp,φ(R1) to hp,φ(R1)? We will give the affirmative answer in section 3 of chapter 2. Our main results are as following:Theorem 0.2.3 Let 0<α≤1< p≤1/1-αand p<∞. If A'∈L∞(R1)∩∧locα(R1), then CA is bounded from Hp,φ(R1) to hp,φ(R1).Compared with Calderon's commutator, the Cauchy integral operator is dif-ficult to study. Because we can calculate TA1(1) and apply "T1 theorem" by David and Journe. But it is difficult to calculate CA(1). To prove the theorem, we have to introduce new methods. Motivated by , we introduce a generalized atom and consider a variant of "Tb theorem": Theorem 0.2.4 Let 0 |
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