A symbolic calculus and L2-boundedness on nilpotent Lie groups

We work on a general nilpotent Lie groupG=G1⊕G2⊕…⊕Gr,where r⩾1 and G(k)=⊕j=kr is the descending central series of G. A composition theorem and an L2 boundedness theorem for convolution operators f→f★A are proved. The composition theorem holds for symbols a=A∧ satisfying the estimates|Dαa(ξ)|⩽Cαm(ξ)g(ξ)−α,where m is a weight andg(ξ)α=g1(ξ)α1…gr(ξ)αr,wheregk(ξ)=1+∑j=k+1r||ξj||21212.The class of weights admissible for the calculus is considerably larger than those of the existing calculi. For the L2-boundedness it is sufficient that|Dαa(ξ)|⩽Cαg(ξ)−α.This goes in the direction of Howeʹs conjecture and improves the results of Howe and Manchon. It is very likely that our methods could also be used to extend the calculus of Melin to general homogeneous groups.