The Boundedness Of Commutators Generated By Riesz Means Of Multiple Hermite Expansions

This paper is concerned with the boundedness of commutator Tλ,bα generated by BMO functions or Lipschitz functions and the Riesz means of multiple Hermi-tian expansions.Firstly, we study the boundedness of commutator Eλ,bα generated by a BMO function and the Riesz means Eλα in Lp(Rn)(n>2) when α> α(p)=max{0, n|1/p-1/2|-1/2}. To deal with the singularity of the commutator, Eλ,bα is decomposed into two parts. One is with singularity Tλ,bα and the other is with non-singularity Rλ,bα. Tλ,bα is decomposed into{Eλ,k,bα}k=0∞·Eλ,0,bα is a commutator of pseudodifferential operators of order zero uniformly when λ is large enough and its boundedness has been already known. To estimate Eλ,k,bα(k≥1), the Euclidean space R2n is divided into S1={(x,y):|x-y|>2kλ1/2} and S2={(x,y):|x-y|≤2kλ1/2}. We use the kernel estimate to compute Eλ,k,bα in S1and use the restriction theorem to estimate Eλ,k,bα in S2.And Rλ,bα is split into the lower frequency part and the higher frequency part. Then we use the restriction theorem to estimate the lower frequency part and kernel estimate to compute the higher frequency part. And by applying Minkowski inequality, the boundedness of Eλ,bα in Lp (Rn) is proved when1<p<pn=2n/(n+2)(n>2) and α>α(p)=max{0, n|1/p-1/2|-1/2}.Secondly, we study the boundedness of commutator Eλ,bα generated by a Lip-schitz function and the Riesz means Eλα using the same technique above and get that Eλ,bα is bounded when1<p<pn=2n/(n+2)(n>2), α> α(p)+1and b∈Λ1.