Two-Weight Norm Estimates For Square Functions Associated To Fractional Schr?dinger Operators With Hardy Potential

In this thesis,we mainly study two-weight norm estimates for square functions associated to fractional Schr?dinger operators with Hardy potential.Let d ∈ N and α∈（0,min{2,d}）.For any a ∈[a*,∞),the fractional Schr?dinger operator L_a is defined by L_a:=（-Δ）^α/2+a|x|^-α,where a*:=-2^αΓ（（d+α）/4）²/Γ(（d-α）/4².First of all,we introduce one kind square functions associated with L_a,and further systematically study the two-weight boundedness of the square function on the weighted Lebesgue space.Then,as an application of these boundedness,we obtain a two-weight Sobolev inequality related to L_a.Finally,we introduce several square functions related to the operator L_a,and obtain the weighted norm estimates for these square functions.The structure of this thesis is as follows:Chapter 1 is Introduction.We mainly introduce the research background and significance of two-weight norm estimates for square functions associated to fractional Schr?dinger operators with Hardy potential,the main results of this thesis,and some common concepts and notations.Chapter 2 is Preliminaries.We introduce some concepts and theorems from harmonic analysis used in this paper,and some existing conclusions that will be used in the proof process of the last two chapters.In Chapter 3,applying some subtle ingredients such as the weighted Hardy inequality associated with the operator L_a,the weighted norm inequalities for the square functions,and a general criterion for two weights boundedness,we establish the twoweight Hardy inequality associated with L_a in the scale of weighted Lebesgue spaces and the two-weight boundedness for square functions associated with L_a in the scale of weighted Lebesgue spaces.In Chapter 4,as applications of off-diagonal estimates,we obtain the weighted norm estimates related to the square functions associated with L_a.In order to prove these estimates,we subtly use the extrapolation theorem and the change of angle formulas.Moreover,we give an application of these estimates of square functions to the Hardy space associated with L_a.In Chapter 5,the conclusions of this thesis are summarized and the future work is prospected.