与薛定谔算子相关的Riesz变换和交换子在加权Morrey空间上的有界性  

Boundedness for Riesz Transforms Related to Schr?dinger Operators and Their Commutators on Weighted Morrey Spaces

作　　者：刘宇[1] 张静[1] 

机构地区：[1]北京科技大学数理学院,北京100083

出　　处：《数学进展》2017年第3期429-440,共12页Advances in Mathematics（China）

基　　金：Supported by NSFC(No.11671031,No.11471018);the Fundamental Research Funds for the Central Universities(No.FRF-TP-14-005C1);Program for New Century Excellent Talents in University;the Beijing Natural Science Foundation(No.1142005)

摘　　要：令L=-△+V是薛定谔算子,其中△是R^n上的拉普拉斯算子,并且非负位势V属于逆H?lder类Bq(q≥n/2).与算子L相关的Riesz变换记为T_1=V(-△+V)^(-1)和T_2=V^(-1/2)(-△+V)^(-1/2),对偶Riesz变换记为T_1~*=(-△+V)^(-1)V和T_2~*=(-△+V)^(-1/2)V^(-1/2).本文建立了T_1~*和T_2~*以及他们的交换子在与位势V∈Bq,q≥n/2相关的加权Morrey空间L_(α,V,ω)^(p,λ)(R^n)上的有界性.这些结果实质性地推广了一些已知的结果.作为应用,本文的结果可以应用于Hermite算子的情形.Let L =-△＋ V be a Schr？dinger operator,where A is the Laplacian on R^n and the nonnegative potential V belongs to the reverse H？lder class Bq for q ≥n/2.The Riesz transforms associated with the operator L are denoted by T1 = V（-△ ＋ V）^（-1） and T2 =V^（-1/2）（-△ ＋ V）^（-1/2） and the dual Riesz transforms are denoted by T1^＊ =（-△ ＋ V）^（-1）V and T2^＊ =（-△ ＋ K）^（-1/2）V^（-1/2）.In this paper,we establish the boundedness for the operator T1^＊ and T2^＊ and their commutators on weighted Morrey spaces L（α,V,ω）^（p,λ）（R^n） associated with the potential V ∈ Bq for q ≥ n/2.These results generalize substantially some well-known results.As an application,we can apply our results to the case of Hermite operators.

关 键 词：MORREY空间 交换子 逆Holder不等式 薛定谔算子 RIESZ变换 

分 类 号：O174.2[理学—数学]