Some Problems On The Limiting Weak-type Behaviors And Compactness For Singular Integrals And Related Operators

In this dissertation,we are devoted to studying the limiting weak-type behaviors for singular integral and related operators and compactness of some commutators.In Chapter One,we summarize the background of the related problems and state the main results of present thesis.We also introduce the main results and give the structure of the dissertation.Chapter Two aims to explore the limiting weak-type behaviors of the Littlewood-Paley functions.Precisely,Under some smooth and size conditions,we establish the limiting weak-type behaviors of the Littlewood-Paley g-functions.Meanwhile,the corresponding results for Marcinkiewicz integral and its fractional version with ker-nels satisfying L?q-Dini condition are also given.In Chapter Three,we study the limiting weak-type behaviors of maximal func-tion associated to power weighted measure.Subsequently,suppose that ??0 and d?(x)=|x|?dx is a power weighted measure on Rn.For 0??<n,we establish the limiting weak-type behaviors of the centered Hardy-Littlewood maximal function and fractional maximal functions associated to measure ?.Meanwhile,the correspond-ing results for the un-centered maximal functions as well as the fractional integral operators with respect to measure ? are also obtained.In Chapter Four,we devote to studying the vector-valued estimates for the lim-iting weak-type behaviors of singular and fractional integrals as well as maximal op-erators with homogeneous kernels.Under the assumptions of that the kernels satisfy certain Dini-type conditions,the limiting weak-type behaviors for the corresponding vector-valued operators are given.Our results imply that the classical vector-valued weak-type endpoint estimates for singular and fractional integrals or maximal opera-tors are not sharp.In Chapter Five,we pay attention to the limiting weak-type behaviors of parabolic singular integral and fractional integral operators with homogeneous kernels satisfying certain Dini condition.Meanwhile,the corresponding results for parabolic maximal singular integral operators and parabolic Marcinkiewicz integral with homogeneous kernels are given.In addition,we also establish the corresponding results for Hardy-Littlewood maximal function on Heisenberg group Hn.In Chapter Six,we focus on the limiting weak-type behaviors related to Riesz transforms and maximal operators associated to Bessel operator.Let ?>0 and??:=-d2/dx2-2?/xd/dx be the Bessel operator on R+:=(0,+?).We establish the limiting weak type behaviors of Riesz transforms associated to ?? Meanwhile,the corresponding results for Hardy-Littlewood maximal operator and fractional max-imal operator in Bessel setting are also obtained.In Chapter Seven,we firstly establish a criterion on the weighted LP compactness of oscillatory Calderon type commutators with p>1,A?Am-1.Meanwhile,for 0<??1,1/p-1/q=?/n,we obtain that oscillatory Calderon type commutators T?A and Calderon type commutators T?A are compact operators from(?p)to Lq(?q)with A ?Am-1,?.Moreover,under certain conditions,the weighted LP compactness of oscillatory Calderon type commutators T?A with rough kernel for A ?m-1,? is also established.