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

摘要

Let L =-△ + V be a Schr(o)dinger operator,where △ is the Laplacian on Rn and the nonnegative potential V belongs to the reverse H(o)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)-1V and T*2 =(-△ + V)-1/2V-1/2.In this paper,we establish the boundedness for the operator T1* and T*2 and their commutators on weighted Morrey spaces Lpα,λV,ω (Rn) 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.%令L=-△+V是薛定谔算子,其中△是Rn上的拉普拉斯算子,并且非负位势V属于逆H(o)1der类Bq(q≥n/2).与算子L相关的Riesz变换记为T1=V(-△+V)-1和T2=V-1/2(-△+V)1/2,对偶Riesz变换记为T1*=(-△+V)-1V和T2*=(-△+V)-1/2V-1/2.本文建立了T1*和T2*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Lp,λα,ν,ω(Rn)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.