Characterizations Of The Boundedness Of Commutators Of Singular Integral Operators

The real-variable theory of BMO space plays an important role in harmonic analy-sis,which has important applications in various fields of analysis and partial differential equations.In this paper,we mainly study the characterizations of the boundedness of commutators of singular integral operators.Then,we answer some open problems affir-matively.In chapter one,the history,development,signification and evolving of the theory of commutator are reviewed.Furthermore,we simply introduce the main work in this paper.In chapter two,characterizations of the weak boundedness of commutators of singu-lar integral operators are considered.We obtain that the commutator of singular integral operator is bounded on weak Lebesgue space if and only if the symbol function belongs to BMO space.In order to find out the reason,the definition of weak central BMO space is introduced and the weak boundedness of Hardy operator is studied.In chapter three,characterizations of BMO space can be obtained by the boundedness of general commutator,linear commutator and iterated commutator of bilinear singular integral operators,respectively.In chapter four,some properties of Campanato spaces are given.We also show that Hardy-Littlewood maximal function is bounded from Lebesgue space to Triebel-Lizorkin space if and only if the symbol function is in Campanato spaces.The final chapter establishes the theory of commutators of Hardy-Littlewood max-imal operators and consider the result in weight Lebesgue spaces and variable exponent Lebesgue spaces.