关键词: Heisenberg群, Folland-Stein定理, 基本解

Abstract: In this note, based on the operator#br#L_a=-1/2∑_j=1ⁿ(Z_jZ_j+Z_jZ_j)+iaT,a∈C,#br#the authors extend consideration to the operator#br#L_λ、μ、α=λ∑_j=1ⁿZ_jZ_j+μ∑_j=1ⁿZ_jZ_j+iaT,λ,μ,α∈C,#br#whereλ+μ≠0且λ≠α/2n,μ≠-α/2n).It is proved that if φ_a,b(z,t)=(|z|²-it)^a(|z|²+it)^b,a=(a-2nλ)/(2(λ+μ)),b=-(a+2nμ)/(2(λ+μ)) such that C_a,b=∫_Hnψ_a,b,1(z,t)dV(z,t)is finite, here ψ_a,b,1(z,t)=-4(λ+μ)ab(|z|²+1-it)^a-1(|z|²+1+it)^b-1,then have#br#L_λ、μ、α φ_a,b=C_a,bδ#br#in the sense of distributions. Especially,as λ=μ=-1/2, the above result is just the Folland Stein theorem.

Key words: Heisenberg group, Folland-Stein theorem, fundamental solution