Multiplier theorem on generalized Heisenberg groups II

Waldemar Hebisch; Jacek Zienkiewicz

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Multiplier theorem on generalized Heisenberg groups

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We study the problem of $L^{p}$ -boundedness ( $1 < p < \infty$ ) of operators of the form $m (L_{1}, \dots, L_{n})$ for a commuting system of self-adjoint left-invariant differential operators $L_{1}, \dots, L_{n}$ on a Lie group $G$ of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when $G$ is a homogeneous group and $L_{1}, \dots, L_{n}$ are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.

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The aim of this paper is to demonstrate how a fairly simple nilpotent Lie algebra can be used as a tool to study differential operators on $ℝ^{n}$ with polynomial coefficients, especially when the property studied depends only on the degree of the polynomials involved and/or the number of variables.

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