A symbolic calculus and L2-boundedness on nilpotent Lie groups

We work on a general nilpotent Lie group [GRAPHICS] where rgreater than or equal to1 and G((k)) = circle plus(j=k)(r) is the descending central series of G. A composition theorem and an L-2 boundedness theorem for convolution operatorsf f-->fstarA are proved. The composition theorem holds for symbols a = A satisfying the estimates [GRAPHICS] where m is a weight and [GRAPHICS] where [GRAPHICS] The class of weights admissible for the calculus is considerably larger than those of the existing calculi. For the L-2-boundedness it is sufficient that [GRAPHICS] This goes in the direction of Howe's 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. (C) 2003 Elsevier Inc. All rights reserved.