Algebras of differential operators on Lie groups and spectral multipliers

TL;DR

This thesis develops a comprehensive theory of spectral multipliers for systems of differential operators on Lie groups, extending classical results to broader classes of groups and operators with applications to harmonic analysis.

Contribution

It introduces new spectral multiplier theorems for systems of differential operators on Lie groups, including non-nilpotent cases, and develops a product theory for multiple groups.

Findings

Proves self-adjointness and strong commutativity of operators under algebraic conditions.

Establishes weighted L^1 estimates for convolution kernels on polynomial growth groups.

Extends Mihlin-H"ormander and Marcinkiewicz multiplier theorems to broader classes of Lie groups.

Abstract

This thesis is devoted to the study of joint spectral multipliers for a system of pairwise commuting, self-adjoint left-invariant differential operators L_1,...,L_n on a connected Lie group G. Under the assumption that the algebra generated by L_1,...,L_n contains a weighted subcoercive operator - a notion due to ter Elst and Robinson (J. Funct. Anal., 157(1):88--163, 1998), including positive elliptic operators, sublaplacians and Rockland operators - we prove that L_1,...,L_n are (essentially) self-adjoint and strongly commuting on L^2(G). Moreover, we perform an abstract study of such a system of operators, in connection with the algebraic structure and the representation theory of G, similarly as what is done in the literature for the algebras of differential operators associated with Gelfand pairs. When G has polynomial volume growth, weighted L^1 estimates are obtained for the convolution kernel of the operator m(L_1,...,L_n) corresponding to a compactly supported multiplier m satisfying some smoothness condition. The order of smoothness which we require on m is related to the degree of polynomial growth of G. In the case G is a homogeneous Lie group and L_1,...,L_n are homogeneous operators, an L^p multiplier theorem of Mihlin-H\"ormander type is proved, extending known results for a single Rockland operator. Further, a product theory is developed, by considering several homogeneous groups G_j, each of which with its own system of operators, and a multiplier theorem of Marcinkiewicz type is proved, not only on the direct product of the G_j, but also on other (possibly non-homogeneous) groups, containing homomorphic images of the G_j. Consequently, for certain non-nilpotent groups of polynomial growth and for some distinguished sublaplacians, we are able to improve the general result of Alexopoulos (Proc. Amer. Math. Soc., 120(3):973-979, 1994).