On The Weighted Boundedness And Compactness For Some Operators

In this dissertation,we mainly consider the weighted boundedness and compactness for singular integrals and related operators.The dissertation is divided into seven chapters.In Chapter One,we sum up the background of the related topics,and formulate the main conclusions of present thesis.Finally,we also present some notations and the structure of this dissertation.In Chapter Two,we focus on the weighted boundedness for the variation and oscillation operators of singular integrals and their commutators.We establish the sparse dominations for the variation and oscillation operators of singular integrals and their commutators with kernels satisfying Hormander condition.As applications,we give the quantitative weighted estimates including the mixed weak type estimates for variation and oscillation operators.Specially,we also get the quantitative weighted weak-type estimates for variation and oscillation operators of commutators,which are new even in the un-weigh ted cases.Besides,We obtain Bloom type estimates for variation and oscillation operators of commutators and a new characterization of weighted BMO spaces.In Chapter Three,we study the weighted compactness of modified oscillation and variation of commutators of singular integrals.We character the CMO spaces via the compactness of modified oscillation and variation of commutators of singular integrals on weighted Lebesgue spaces.Meanwhile,we give a counterexample to illustrate the classical oscillation of commutator of singular integral is not enough to character the CMO spaces,and the necessity of this variant of oscillation.In Chapter Four,we pay attention to study the boundedness and compactness of commutators for maximal operators with rough kernels.We give the characterizations of BMOα and CMOa spaces via the boundedness and compactness for commutators of maximal operators with rough kernels in weighted Lebesgue spaces,respectively.As a special case,we obtain characterized theorems on the boundedness and compactness for commutators of Hardy-Littlewood maximal operator and fractional maximal operator in weighted Lebesgue spaces,which are new even in the un-weighted cases.In Chapter Five,we study the boundedness and compactness of commutators for Marcinkiewicz integral with rough kernels,we establish the characterizations of BMOα and CMOα,spaces via the boundedness and compactness for commutators of Marcinkiewicz integrals with rough kernels in weighted Lebesgue spaces,respectively,which improve the previous results.Meanwhile,we also give the quantitative weighted bounds for the commutator of Marcinkiewicz integral.In Chapter Six,we investigate the weighted boundedness of multilinear pseudodifferential operators and their commutators,we establish the sparse dominations for the iterated commutators of multilinear pseudo-differential operators with the symbol belonging to the Hormander class.Based on this dominations,we obtain the quantitative bounds of the Bloom type estimates for the iterated commutators of multilinear pseudo-differential operators with the symbol belonging to the Hormander class.Moreover,we also obtain the Cp estimates for the corresponding multilinear pseudodifferential operators.Chapter Seven aims to investigate the boundedness of maximal operators of multilinear singular integrals on weighted Hardy spaces and variable Hardy spaces.We show that the maximal operators of the multilinear Calderón-Zygmund operators are bounded from a product of weighted Hardy spaces into weighted Lebesgue spaces,which greatly extend and improve the previous known results.Furthermore,we also obtain the corresponding estimates on variable Hardy spaces.