Arkiv för Matematik

The Melin calculus for general homogeneous groups

Paweł Głowacki

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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 ρj are “partial homogeneous norms” depending on the variables ξk 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.

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Ark. Mat., Volume 45, Number 1 (2007), 31-48.

Received: 17 October 2005
First available in Project Euclid: 31 January 2017

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2007 © Institut Mittag-Leffler


Głowacki, Paweł. The Melin calculus for general homogeneous groups. Ark. Mat. 45 (2007), no. 1, 31--48. doi:10.1007/s11512-006-0034-5.

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