Some Operators And Commutators On Function Spaces

This PhD thesis focuses on the boundedness of some important integral op-erators and their commutators on function spaces with both constant and vari-able exponents. The operators we mainly consider here are the high-dimensional Hausdorff, the fractional Hausdorff, the multilinear Hausdorff and multilinear sin-gular integrals with smooth kernels. Our theorems here can be regarded as the extension of some known results in the literature. The boundedness of Hausdorff operators and their commutators on function spaces with constant exponents covers the main part of the thesis. However, multilinear singular integrals along with their commutators on Herz-type spaces with variable exponents are also studied. In addition, the boundedness of Hardy operators has also been obtained as a special case of Hausdorff operators. Compared with Hardy operators, the boundedness of Hausdorff operators is far from perfect, therefore, more additional techniques have been presented in the development of this thesis.In the first part of this dissertation, we give estimates for Hausdorff operators and their commutators with central BMO functions or Lipschitz functions on function spaces with constant exponents including Herz, Morrey-Herz and central Morrey spaces. However, the second part is devoted to study the boundedness of multilinear singular integrals and commutators generated by these operators and BMO functions on variable exponent Herz-type spaces. On the basis of such motivations we divide the whole thesis into five chapters. A brief summary of the results contained in each of these chapters is as follows:Chapter1is introduction and contains some basic information about the operators, their historical background and definitions of some important function spaces. In this Chapter, we give a brief review of Hausdorff operators and their recent developments. Similar analysis for singular integrals is also provided. Furthermore, some function spaces with constant and variable exponents are introduced briefly.In Chapter2, we mainly study boundedness of high-dimensional Hausdorff operator recently introduced by Chen,Fan and Li in[6],and obtain sharp bounds for HΦ on some function spaces. Thus,our main theorems regarding such bounds for HΦ are: Theorem2.2.1.Let α∈R,λ≥0,1<p,q<∞.If Φ is a non-negative valued function and then HΦ is a bounded operator on Herz-Morrey space MKp,qα,λ(Rn).Conversely, suppose that HΦ is a bounded operator on MKp,qα,λ(Rn).Ifλ=0, or if λ>max{0,κ},then A1<∞.In addition,the operator HΦ satisfies the following operator norm‖HΦ‖MKp,qα,λ(Rn)→MKp,qα,λ(Rn)=A1. Theorem2.2.2.Let-1/p≤λ<0,1<p<∞.If Φ is a non-negative valued function and then HΦ is a bounded operator on central Morrey space Bp,λ(Rn).Conversely,suppose that HΦ is a bounded operator on Bp,λ(Rn).If λ=-1/p or if-1/p<λ<0,then A2<∞.Furthermore,the operator HΦ satisfies the following operator norm‖HΦ‖Bp,λ(Rn)→Bp,λ(Rn)=A2.In addition,we introduce the commutators HΦb=bHΦ-HΦb of HΦ with central BMO or Lipschitz functions b and give following estimates: Theorem2.3.1.Let b∈Λβ(Rn),0<β<1<q2<q1<∞,0<p<∞,η=α+β+n/q2-n/q1.If then HΦb is bounded from Mp,q1η,λ(Rn) to MKp,q2α,λ(Rn) and satisfies the following inequality Theorem2.3.2. Let b∈CMOr(Rn),r=q1q2/q1-q2,1<q2<q1<∞,0<p<∞,θ=α+n/q2-n/q1. If then HΦb is bounded from Mkp,q1θ,λ(Rn) to MKp,q2α,λ(Rn) and satisfies the following inequalityIt is worth mentioning here that such estimate have never been reported in the literature and yield many corollaries as special cases (see Chapter2).The theory of Hausdorff operator is in the process of development and new concepts are being introduced with the passage of time. As a part of this devel-opment Sun and Lin [52] have introduced the fractional Hausdorff operator given by which is a modified form of the operator studied in [6]. The commutators HΦ,γb=bHΦ,γ-HΦ,γb of HΦ,γ with central BMO or Lipschitz functions b are discussed in Chapter3. In addition central BMO estimates for commutators HΦ,b=bHΦ-HΦb of HΦ on central Morrey space are also obtained. The generality of these estimate is such that we can deduce similar estimates for Hardy operators as special cases. Below are our main theorems of Chapter3. Theorem3.2.1. Let then HΦ,γb is bounded from MKq,q1μ,λ(Rn) to MKp,q2α,λ(Rn) and satisfies the following inequality Theorem3.3.1. Let then HΦ,γb is bounded from MKq,q1v,λ(Rn) to MKp,q2α,λ(Rn) and satisfies the following inequality Theorem3.4.1. Let1<p1<∞, p1’<p2<∞,1/p=1/p1+1/p2,-1/p<λ<0, and b∈CMOp2(Rn). If where Φ is a radial function on (0,∞). Then the commutator HΦ,b is bounded from Bp1,λ to Bp,λ and satisfies the following inequality‖HΦ,bf‖Bp,λ(Rn)≤CD5‖b‖CMOp2(Rn)‖f‖Bp1,λ(Rn).We refer the reader to the paper [24] for further results regarding the bound-edness of HΦ,b.The necessary part of the so-called developing theory of Hausdorff operators is its multilinear version. In [7], Chen, Fan and Zhang, recently introduced multilinear extensions of n-dimensional Hausdorff operator. One of them is the operator The second multilinear extension is given by For locally integrable functions b1and b2, we define the commutators of2-linear Hausdorff operator SΨ(f1,f2) to be[b1,b2,SΨ](f1,f2)(x)＝b1(x)b2(x)SΨR(f1,f2)(x)-b1(x)SΨ(f1,b2f2)(x)-b2(x)SΨ(b1f1,f2)(x)+SΨ(b1f1,b2f2)(x).The main goal of Chapter4is to study the boundedness properties of mul-tilinear Hausdorff operators and their commutators on central Morrey and Herz space, respectively. As special cases of our results boundedness of multilinear Hardy operators is achieved which will confirm some already obtained results. Here, we state the main results of this Chapter. Theorem4.2.1. Let m∈N,fi be in Bpi,λi(Rn), If the radial function Φ satisfies thenTheorem4.2.2. Let SΨ be defined as above, and fi∈Bpi,λi. If then Theorem4.3.1. Let then [b1,b2,SΨ] is bounded from Kq1η1,l1×Kq2η2,l2to Kpα,l and satisfy the following inequalityIn2002, Grafakos and Torres proved that the multilinear singular integralT(f1,…,fm)(x)=∫(rn)mK(x,y1,…,ym)f1(y1)…fm(ym)dy1…dym,where K is m-linear Calderon-Zygmund kernal, is bounded operator on product of Lebesgue spaces and endpoint weak estimates hold, see [29]. For suitable function f and g, Huang and Xu [34] defined the commutators of T with BMO functions b1and b2as [bb,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 Lebesgue space. In Chapter5, we will consider the problem of boundedness of multilinear singular integral operator T on variable exponent Herz-type spaces which can be considered as extensions of some results provided in [34]. The commutators of T with BMO functions are also discussed. Thus the main results are: Theorem5.2.1. Let T be2-linear Calderon-Zygmund operator and p(·)∈P(Rn). Furthermore, let p1(·),p2(·)∈B(Rn),0<qt<∞,-nδpl<α1<nδpl’, l=1,2, where δpl,δpl’>0are constants, such thatα=α1+α2.If T is bounded from Lp1(·)×Lp2(·) to Lp(·),then‖T(f,g)‖Kp(·)α,q≤C‖f‖Kp1(·)α1,q1‖g‖Kp2(·)α2,q2and hold for all f, fh∈Kp1(·)α1,q1, g, gh∈Kp2(·)α2,q2, where1<rl<∞for l=1,2and1/r=1/r1+1/r2. Theorem5.2.2.Let T be2-linear Calderon-Zygmund operator,b1,b2∈BMO(Rn) and p(·)∈P(Rn).Furthermore,let p1(·),p2(·)∈B(Rn),0<q1<∞,-nδpl<αl<nδpl’,l=1,2,where δp1,δpl’>0are constants,such that α=α1+α2.If[b1,b2,T]is bounded from Lp1(·)×Lp22(·) to Lp(·),then‖[b1,b2,T](f,g)‖Kp(·)α,q≤G‖b1‖*‖b2‖*‖f‖Kp1(·)α1,q1‖g‖Kp2(·)α2,q2and hold for allIt is to be noted that our results not only extend some theorems in[34] but also generalize some results in [69] and [80].