Boundedness Of Commutators Of Singular Integral Operators

It is well known that the singular integrals and its'commutators play a profound and extensive role in harmonic analysis and the partial differential equations. In this thesis, we will discuss the properties of this kind of operators and its'commutators. This thesis consists of four chapters. Chapter I is the introduction. Chapter II, Chapter III and Chapter IV are the main contents of this thesis.Chapter I is introduction.In Chapter II, we discuss the boundedness of the commutators of generalized Calderon-Zygmund operators in Hardy type spaces, and two parts of conclusions are obtained. In first part, by Minkowski integral inequality and Jensen inequality seldom used before to control some inequalities, we studied the character of commutators generated by generalized Calderon-Zygmund operators and Lips-chitz function, and obtained the boundedness of the commutator generated by 0(?-type Calderon-Zygmund operator T and Lipschitz function b on Hardy spaces and Herz type Hardy spaces. In second part, we proved that this commutator is bounded from Hardy spaces to weak Lebesgue spaces and from Herz type Hardy spaces to weak Herz spaces on critical point. The backgrounds of the two questions are as following: (t)-type Calder n-Zygmund operator was introduced in [27] by Yabuta and in [20] by Peng Lizhong. As to some studies of (t)- type Calderon-Zygmund operator (we can see [27], [20], [28], [29], [30] etc for details). In 2002, Lu Shanzhen, Wu Qiang and Yang Dachun studied the boundedness of commutators generated by standard Calderon-Zygmund operators and Lipschitz functions on Hardy type spaces in [15].Inspired by the results in [15], we obtain the following conclusions:Theorem 2.2 Let b Lip (Rn)(0 < < 1), and dt < + ,then [b,T] is bounded from Theorem 2.3 Let b Lip (Rn)(0 < < 1). If 0 < p < + ,l < q1q2 < + , 1/q2 = dt < , then [b, T] is boundedfrom HK (Rn) to K (Rn).Theorem 2.4 b Lip (Rn)(0 < 1). If 0 < 1,1 < q1,q2 < + , 1/q2 =In Chapter III, we discuss the commutator [g, Tb] generated by strongly singular integral operator Tb and BMO function g. In 1998, Li Xiaochun and Lu Shanzhen discussed the strongly singular integral operators in weighted Herz type Hardy spaces in [13], In this Chapter, we extend the conclusion of [13] to the commutator [g,7b], It is proved that the commutators generated by strongly singular integral operators and BMO functions are bounded from the homogeneous weighted Herz-type Hardy spaces HK ( 1, 2) to the homogeneous weighted Herz spaces K( 1, 2) when = n(l - 1/q). The conclusions are as following:Theorem 3.1 Let 0 < p 1< q < , = n(l -1/q), g BMO(Rn), and 1, 2 A1, thenhere C is a constant independent of /.In Chapter IV, It is proved that commutator [b, T] generated by BMO function 6 and generalized Calderon-Zygmund operator T is bounded on Herz type Hardy spaces and weighted Herz type Hardy spaces. In 2001, Liu Zongguang discussed the boundedness of commutator [b,T] of standard singular integral operators in [14]. Inspired by the results in [14], we obtain the following conclusions:Theorem 4.1 Let b 6 BMO(jRn),0 < p < oo,l < q < oo, there is e 6 (0, 1] such that n(l - l/g)