Algebras of differential operators on Lie groups and spectral multipliers
Alessio Martini

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…
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Taxonomy
TopicsAdvanced Harmonic Analysis Research · Advanced Mathematical Physics Problems · Advanced Algebra and Geometry
