Spectral multipliers on nilpotent Lie groups and homogeneous spaces

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For a system of strongly commuting self-adjoint operators on some L^2-space, a joint functional calculus is defined via spectral integration, where bounded Borel functions correspond to L^2-bounded operators. L^p-boundedness (for other values of p) of such operators is much more difficult to characterize in terms of the defining functions, called spectral multipliers. Several problems and results of harmonic analysis fall into this frame, which involve various techniques such as the Calderon-Zygmund theory of singular integral operators, and have applications in the study of partial differential equations. This project is mainly concerned with the case of a system of group-invariant differential operators on a nilpotent Lie group. Although quite general theorems giving sufficient conditions for the L^p-boundedness in terms of smoothness of the multiplier are available, only in few cases sharp results have been proved, requiring a detailed knowledge of the underlying algebraic structure and highlighting a nontrivial interaction between Euclidean and sub-Riemannian geometric structures. Recently a new technique has been developed to deal with the case of a homogeneous sublaplacian on a 2-step group. The aim of this project is to extend the new technique and to obtain sharp L^p-boundedness results for wider classes of groups and operators, and for systems of commuting operators.