Two-Weight Inequalities For Several Type Of Integral Operators

Researching weighted inequalities for Hardy–Littlewood maximal operator, singularintegral operators, and fractional integral operators and other operators is an importanttopics in harmonic analysis. The main research content is to explore the conditions thatweighted inequalities hold for these integral operators on anisotropic function spaces inharmonic analysis, and harmonic analysis has been applied widely to partial di?erentialequations, approximation theory of functions and other fields.In 1972, the research of weighted theory of harmonic analysis began by Muckenhoupt.He studied the conditions of weighted functions such that Hardy–Littlewood maximaloperator is bounded on Lebesgue spaces, and established weighted theory of ????. LaterCoifman and Fe?erman gave the weighted inequalities for singular integral operators.Muckenhoupt, Wheeden and Sawer also obtained the corresponding weighted results forfractional integral operators and fractional maximal operators, respectively.Two-weight problems originated from studying two-weight inequalities for Hardy–Littlewood maximal operator ?? on Lebesgue spaces, that is, the conditions of weightfunctions(??, ??) such that the operator ?? : ????(??) â†' ????(??)(1 < ?? < ∞) is bounded arediscussed. It is the weighted problem studied by Muckenhoupt when ?? = ??. Fe?ermanand Stein proved that Hardy–Littlewood maximal operator ?? satisfies ‖?? ?? ‖????(??)≤??‖?? ‖????(?? ??)for any weighted function, that is, ?? satisfies two-weight inequalities for anyweight. Sawyer gave the necessary and su?cient conditions of Sawyer type for the weight-ed function(??, ??) such that the maximal operator ?? satisfies two-weight inequalities.Since 1994, Perez and Cruz-Uribe studied two-weight inequalities of Hardy–Littlewoodmaximal operator, singular integral operators and their commutators, fractional inte-gral operators, and so on, and obtained the su?cient conditions of ????type such thattwo-weight strong-type or weak-type norm inequalities of these operators hold.The weighted theory has been expanded to study weighted inequalities of other in-tegral operators and multilinear operators. In addition, studying for characteristics ofweight functions has made many achievements now, and has developed the theory ofweighted interpolation, weighted extrapolation, and so on. The weighted theory is es-tablished on homogeneous spaces and non-homogeneous spaces. Because these spacesare general, weighted theory is made more widely used. Previously, people had a majorresearch on weighted inequalities of kinds of integral operators on Lebesgue spaces. Re-cently, these theories have been expanded on Lorentz spaces ????,??, Morrey spaces ????,??,and so on, but few results have been obtained.Many achievements have been made in two-weight theory of integral operators now,but there are many problems to be studied. This dissertation discussed two-weight in-equalities for Hardy operator, two-weight extrapolation on Lorentz Spaces, two-weightinequalities for fractional maximal operators on Morrey spaces, two-weight inequalitiesfor multilinear fractional maximal operators and multilinear fractional integral operatorson non-homogeneous spaces, and so on.The paper is divided into four chapters.In chapter 1, two-weight boundedness of Hardy operator ?? and the related op-erators on(0, ∞) are discussed. Firstly, for the maximal operator ?? related to theHardy operator, we obtained the necessary and su?cient conditions of ????type such that?? : ????(??) â†' ????,∞(??) is bounded and the necessary and su?cient conditions of Sawyertype such that ?? : ????(??) â†' ????(??) is bounded, respectively. These results extend theresults of Sawyer’s two-weight inequalities for Hardy–Littlewood maximal operator. Wealso obtain ????type su?cient conditions of the two-weight inequalities for Hardy operator??, its adjoint operator ?? and the commutators of these operators with CMO functions.Hardy operator is a basic and important operator in function theory, and there arelots of research results about them, and lots of weighted inequalities for integral operatorshave been widely applied in the analysis. What we have obtained are quite di?erent fromthat in the past.In chapter 2, several two-weight extrapolation theorems on Lorentz spaces are given,as their applications, two-weight inequalities about any weight for Hardy–Littlewoodmaximal operator, singular integral operators and commutators on Lorentz spaces areobtained.The classical extrapolation theorem is due to Rubio de Francia. Cruz-Uribe, Martell,P′erez, et al generalized the extrapolation theorem of Rubio de Francia and got extrapola-tion theorems associated to ??∞ weights. These theorems have been proved to be the keyto solving many problems in harmonic analysis. Cruz-Uribe and P′erez gave extrapola-tion theorems for pairs of weights of the form(??, ??????) and(??,(?? ??/??)????) on Lebesguespaces, we extend the two-weight extrapolation theorems of Cruz-Uribe and P′erez onLorentz spaces in this chapter.In chapter 3, some su?cient conditions and necessary conditions such that the two-weight inequalities hold for fractional maximal operators on Morrey spaces are discussed.Many results about weighted inequalities for fractional maximal operators have beenobtained on Lebesgue spaces. Two-weighted inequalities for the Hardy-Littlewood maxi-mal operator on weighted Morrey spaces were discussed by Ye and Wang, and Sawyer typesu?cient conditions are obtained. in this chapter we expand these results to fractionalmaximal operators. In addition, we obtained ????type su?cient conditions.In chapter 4, Two-weight strong-type inequalities for multilinear fractional maximaloperators and multilinear fractional integral operators on non-homogeneous spaces aregiven, and Sawyer type su?cient conditions and ????su?cient type conditions such thattwo-weight strong-type inequalities hold are obtained.The multilinear maximal operators ? were introduced by Lerner, Ombrosi, P′erez,Torres and Trujillo-Gonz′alez, and the multilinear fractional maximal operator ???wereintroduced by Moen. They are the multilinear extension of Hardy–Littlewood maxi-mal operator and fractional maximal operators. Li, Xue and Yan gave the two-weightinequalities for ???, and obtained Sawyer type su?cient conditions.Because the measure on non-homogeneous spaces doesn’t satisfy double conditionand only satisfies growth condition, which bring much trouble to the research, the resultsof two-weight inequalities on non-homogeneous spaces are few. Garc′?a-Cuerva and Martellgave two-weight inequalities for fractional maximal operators on non-homogeneous spaces,in this chapter, we expanded their results.