Analysis of joint spectral multipliers on Lie groups of polynomial growth

TL;DR

This paper investigates the boundedness of joint spectral multipliers on Lie groups of polynomial growth, extending classical multiplier theorems to a broader class of non-commutative groups with specific differential operators.

Contribution

It establishes L^p-boundedness results for spectral multipliers on Lie groups of polynomial growth, including analogues of Mihlin-H"ormander and Marcinkiewicz theorems for homogeneous groups.

Findings

Proved L^p-boundedness of spectral multipliers for certain Lie groups.

Extended classical multiplier theorems to non-commutative settings.

Established conditions under which operators are bounded on L^p spaces.

Abstract

We study the problem of L^p-boundedness (1 < p < \infty) of operators of the form m(L_1,...,L_n) for a commuting system of self-adjoint left-invariant differential operators L_1,...,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,...,L_n are homogeneous, we prove analogues of the Mihlin-H\"ormander and Marcinkiewicz multiplier theorems.