The Melin calculus for general homogeneous groups

Abstract

The purpose of this note is to give an extension of the symbolic calculus of Melin for convolution operators on nilpotent Lie groups with dilations. Whereas the calculus of Melin is restricted to stratified nilpotent groups, the extension offered here is valid for general homogeneous groups. Another improvement concerns the L²-boundedness theorem, where our assumptions on the symbol are relaxed. The zero-class conditions that we require are of the type $|D^{\alpha}a(\xi)|\le C_{\alpha}\prod_{j=1}^R\rho_j(\xi)^{-|\alpha_j|},$ where ρ_j are “partial homogeneous norms” depending on the variables ξ_k for k> j in the natural grading of the Lie algebra (and its dual) determined by dilations. Finally, the class of admissible weights for our calculus is substantially broader. Let us also emphasize the relative simplicity of our argument compared to that of Melin.