Some Operators and Commutators on Function Spaces

摘要

This PhD thesis focuses on the boundedness of some important integral operators and their commutators on function spaces with both constant and variable exponents.The operators we mainly consider here are the high-dimensionalHausdorff, the fractional Hausdorff, the multilinear Hausdorff and multilinear singular integrals with smooth kernels.Our theorems here can be regarded as theextension of some known results in the literature.The boundedness of Hausdorffoperators and their commutators on function spaces with constant exponentscovers the main part of the thesis.However, multilinear singular integrals alongwith their commutators on Herz-type spaces with variable exponents are alsostudied.In addition, the boundedness of Hardy operators has also been obtainedas a special case of Hausdorff operators.Compared with Hardy operators, theboundedness of Hausdorff operators is far from perfect, therefore, more additionaltechniques have been presented in the development of this thesis.
　　In the first part of this dissertation, we give estimates for Hausdorff operatorsand their commutators with central BMO functions or Lipschitz functions onfunction spaces with constant exponents including Herz, Morrey-Herz and centralMorrey spaces.However, the second part is devoted to study the boundednessof multilinear singular integrals and commutators generated by these operatorsand BMO functions on variable exponent Herz-type spaces.On the basis of suchmotivations we divide the whole thesis into five chapters.A brief summary ofthe results contained in each of these chapters is as follows:
　　Chapter 1 is introduction and contains some basic information about theoperators, their historical background and definitions of some important functionspaces.In this Chapter, we give a brief review of Hausdorff operators and theirrecent developments.Similar analysis for singular integrals is also provided.Furthermore, some function spaces with constant and variable exponents areintroduced briefly.
　　In Chapter 2, we mainly study boundedness of high-dimensional HausdorffoperatorHφf(x) =∫Rnφ(y)/|y|nf(x/|y|)dy,recently introduced by Chen, Fan and Li in [6], and obtain sharp bounds for Hφon some function spaces.Thus, our main theorems regarding such bounds forHφ are:
　　Theorem 2.2.1.Let α ∈ R, λ ≥ 0, 1 ＜ p, q ＜ ∞.If φ is a non-negative valuedfunction andA1=∫Rnφ(y)/|y|n|y|α+n/q-λdy ＜∞，then Hφ is a bounded operator on Herz-Morrey space M (K)α,λp,q(Rn).
　　Conversely, suppose that Hφ is a bounded operator on M (K)α,λp,q(Rn).If λ =0,or if λ ＞ max{0,α}, then A1 ＜ ∞.In addition, the operator Hφ satisfies thefollowing operator norm‖Hφ‖M (K)α,λp,q(Rn)→M (K)α,λp,q(Rn)=A1.
　　Theorem 2.2.2.Let-1/p ≤ λ＜ 0, 1 ＜ p ＜ ∞.If φ is a non-negative valuedfunction andA2 =∫Rnφ(y)/|y|n|y|-nλdy ＜ ∞,then Hφ is a bounded operator on central Morrey space (B)p,λ(Rn).
　　Conversely, suppose that Hφ is a bounded operator on (B)p,λ(Rn).If λ =-1/por if-1/p ＜ λ ＜ 0, then A2 ＜ ∞.Furthermore, the operator Hφ satisfies thefollowing operator norm‖Hφ‖(B)p, λ(Rn)→(B)p, λ(Rn)=A2.
　　In addition, we introduce the commutators Hbφ =bHφ-Hφb of Hφ withcentral BMO or Lipschitz functions b and give following estimates:
　　Theorem 2.3.1.Let b ∈ (∧) β(Rn),0 ＜ β ＜ 1 ＜ q2 ＜ q1 ＜ ∞,0 ＜ p ＜∞,η =α+β+ n/q2-n/q1.IfA3 =∫Rn|φ(y)|/|y|n|y|α+n/q2-λ max{1, |y|β}dy ＜ ∞,then Hbφ is bounded from M (K)η,λp,q1(Rn) to M (K)α,λp,q2(Rn) and satisfies the followininequality‖Hbφ,f‖M (K)α,λp,q2(Rn)≤ CA3‖b‖(∧)β(Rn)‖f‖M(K)η,λp,q1(Rn).
　　Theorem 2.3.2.Let b ∈ C(M)Or(Rn),r =q1q2/q1-q2, 1 ＜ q2 ＜ q1 ＜∞,0 ＜ p ＜∞,θ=α+n/q2-n/q1.IfA6 =∫Rn |φ(y)|/|y|n|y|α+n/q2-λ(2+logmax{|y|,1/|y|})dy ＜ ∞,then Hb is bounded from M(K)θ,λp,q1(Rn)to M (K)α,λp,q2(Rn) and satisfies the following)inequality‖Hbφ f‖MKα,λp,q2(Rn) ≤ CA6‖b‖C(M)Or(Rn)‖f‖M(K)θ,λp,q1(Rn).
　　It is worth mentioning here that such estimate have never been reported inthe literature and yield many corollaries as special cases (see Chapter 2). The theory of Hausdorff operator is in the process of development and newconcepts are being introduced with the passage of time.As a part of this development Sun and Lin [52] have introduced the fractional Hausdorff operator givenbyHφ,γf(x)=∫Rnφ(|x|/|y|)/|y|n-λf(y)dy.which is a modified form of the operator(H)φf(x) =∫Rn φ(|x|/|y|)/|y|nf(y)dy,studied in [6].The commutators Hbφ,γ =bHφ,γ-Hφ,γb of Hφ,γ with central BMOor Lipschitz functions b are discussed in Chapter 3.In addition central BMOestimates for commutators (H)φ,b =b(H)φ-(H)φb of (H)φ on central Morrey space arealso obtained.The generality of these estimate is such that we can deduce similarestimates for Hardy operators as special cases.Below are our main theorems ofChapter 3.
　　Theorem 3.2.1.Let b ∈ (∧)β(Rn), 0 ＜ β ＜ 1 ＜ q2 ＜ q1 ＜ ∞, 0 ＜ p ＜∞， λ ＞ 0,μ=α+β+γ+n/q2-n/q1.IfD1 =∫∞0 |φ(t)|ttα+n/q2-λmax{1, tβ}dt ＜ ∞,then Hbφ,γ is bounded from M (K)μ,λp,q1(Rn) to M (K)α,λp,q2(Rn) and satisfies the followinginequality‖Hbφ,γf‖M (K)α,λp,q2(Rn)≤ CD1‖b‖(∧)β(Rn)‖f‖M(K)μ,λp,q1(Rn).
　　Theorem 3.3.1.Let b∈ CMOr(Rn),r =q1q2/q1-q2, 1 ＜ q2 ＜ q1 ＜ ∞, 0 ＜ p ＜∞, v =α+γ+ n/q2-n/q1.IfD3 =∫∞0 |φ(t)|/ttα+n/q2-λ(2+logmax{t,1/t})dy ＜ ∞,then Hbφ,γ is bounded from M (K)v,λp,q1(Rn) to M (K)α,λp,q2(Rn) and satisfies the followinginequality‖Hbφ,γf‖M (K)α,λp,q2(Rn)≤ CD3‖b‖C(M)Or(Rn)‖f‖M(K)μ,λp,q1(Rn).
　　Theorem 3.4.1.Let 1 ＜ p1 ＜ ∞, p'1 ＜ p2 ＜ ∞, 1/p =1/p1+1/p2,-1/p＜λ＜0,andb∈ C(M)Op2(Rn) IfD5 =∫∞0|φ(t)|/t1+nλ(2 + log max{t,1/t}) dt ＜ ∞,whereΦ is a radial function on (0,∞).Then the commutator (H)φ,b is boundedfrom (B)p1,λ to (B)p，λ and satisfies the following inequality‖(H)φ,bf‖(B)p,λ(Rn)≤ CD5‖b‖C(M)Op2(Rn)‖f‖(B)p1,λ(Rn).
　　We refer the reader to the paper [24] for further results regarding the bound-edness of (H)φ,b.
　　The necessary part of the so-called developing theory of Hausdorff operatorsis its multilinear version.In [7], Chen, Fan and Zhang, recently introducedmultilinear extensions of n-dimensional Hausdorff operator.One of them is theoperatorTφ(f1,…,fm)(x)=∫Rnmφ(x/|y|)/|y|nmm∏i=1fi(yi)dy.The second multilinear extension is given bySψ(f1,…,fm)=∫Rnmψ(x/|y1|,…,x/|ym|)/|y1|n…|ym|nm∏i=1fi(yi)dy1…dym.For locally integrable functions b1 and b2, we define the commutators of 2-linearHausdorff operator Sψ(f1, f2) to be[b1, b2, Sψ](f1, f2) (x) =b1 (x)b2(x)Sψ (f1, f2)(x)-b1(x)Sψ (f1, b2 f2)(x)-b2(x)Sψ (b1f1, f2)(x) + Sψ(b1f1, b2f2)(x). The main goal of Chapter 4 is to study the boundedness properties of multilinear Hausdorff operators and their commutators on central Morrey and Herzspace, respectively.As special cases of our results boundedness of multilinearHardy operators is achieved which will confirm some already obtained results.Here, we state the main results of this Chapter. Theorem 4.2.1.Let m ∈ N, fi be in (B)pi,λi(Rn), 1 ＞ 1/p=1/p1+ …1/pm,-1/pi≤λi ＜ 0, i =1, …, m, and λ =λ1 + … +λm.If the radial function φ satisfiesK3 =∫Bnm|φ(1/|y|)/|y|nmm ∏i=1|yi|nλidy＜∞,then‖Tφ(f1, …, fm)（B）p，λ ≤K3m∏i=1‖fi‖（B）pi,λi.
　　Theorem 4.2.2.Let Sψ be defined as above, 1 ＞ 1/p =1/p1+1/p2,-1/pi≤ λi ＜0, i =1, 2,λ =λ1 + λ2, and fi ∈ （B）λi,pi.IfK4 =∫∞0∫∞0 |Φ(t1,t2)|/t11+nλ1t12+λ2dt1dt2 ＜∞,then‖Sψ(f1,f2)‖(B)p,λ≤ |Sn-1|2K42∏i=1‖fi‖(B)pi,λi.
　　Theorem 4.3.1.Let 1 ＜ qi, li ＜ ∞, bi ∈ (∧)βi,0 ＜βi ＜ 1,1/q =∑2i=11/qi,1/i=∑2i=1 1/li,1/r=∑2i=1 1/ri,ηi=αi+βi+n/ri, 1/p=1/q+1/r.IfK5 =∫∞0∫∞0|ψ(t1,t2)|/t11-α1-n/q1-n/r1t12-α2-n/q2-n/r22∏i=1 max{1, tβii}dt1dt2＜∞,then [b1, b2, Sψ] is bounded from (K)η1,l1q1 ×(K)η2,l2q2 to (K)α,lp and satisfy the followinginequality‖[b1,b2,Sψ](f1,f2)‖(K)α,lp ≤ CK5 2∏i=1 (‖bi‖(∧)βi‖fi‖(K)η2,l2qi).
　　In 2002, Grafakos and Torres proved that the multilinear singular integralT(f1, …, fm)(x) =∫(Rn)m K(x, y1,…, ym)f1(y1)…fm(ym)dy1…dym,where K is m-linear Calderón-Zygmund kernal, is bounded operator on productof Lebesgue spaces and endpoint weak estimates hold, see [29].For suitablefunction f and g, Huang and Xu [34] defined the commutators of T with BMOfunctions b1 and b2 as[b1, b2, T](f , g)(x) =b1(x)b2(x)T(f , g)(x)-b1(x)T(f, b2g)(x)-b2(x)T(b1f, g)(x)+ T(b1f, b2g)(x),and proved the boundedness of both T and [b1, b2, T] on variable exponent Lebesguespace.In Chapter 5, we will consider the problem of boundedness of multilinearsingular integral operator T on variable exponent Herz-type spaces which can beconsidered as extensions of some results provided in [34].The commutators of Twith BMO functions are also discussed.Thus the main results are:

目录

Acknowledgement

Abstract

Chapter 1 Introduction and Preliminaries

1．1 Introduction

1．2 Some Function Spaces

1．3 Generalized Function Spaces

Chapter 2 On Multidimensional Hausdorff Operator HΦ

2．1 Introduction

2．2 Sharp Bounds for HΦ

2．2．1 Main Results

2．2．2 Proofs of Main Results

2．3 The Commutators of HΦ

2．3．1 Main Results

2．3．2 Preliminary Lammas

2．3．3 Proofs of Main Results

Chapter 3 Commutators of Fractional Hausdorff Operators

3．1 Introduction

3．2 Lipschitz Estimates on Herz-Type Spaces

3．3 Central BMO Estimates on Herz-Type Spaces

3．4 Central BMO Estimates on Central Morrey Space

Chapter 4 Multilinear Hausdorff Operators and Commutators

4．1 Introduction

4．2 Multilinear Hausdorff Operators on Central Morrey Space

4．3 Commutators of Multilinear Hausdorff Operator on Herz Space

Chapter 5 Singular Integrals on Variable Exponent Herz-type Spaces

5．1 Introduction

5．2 Boundedness on Variable Exponent Herz Space

5．2．1 Main Results

5．2．2 Preliminary Lemmas

5．2．3 Proofs of Main Results

5．3 Boundedness on Variable Exponent Morrey-Herz Space

5．3．1 Main Results

5．3．2 Proofs of Main Results

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