@PhdThesis{macau_mods_00004413, author = {Niedorf, Lars}, title = {Spectral multipliers on two-step structures}, year = {2024}, publisher = {Christian-Albrechts-Universit{\"a}t zu Kiel}, address = {Kiel}, keywords = {spectral multiplier; two-step structure; sub-Riemannian geometry; Grushin operator; Lie group; sub-Laplacian; restriction type estimate}, abstract = {In harmonic analysis and partial differential equations, the topics of Fourier restriction estimates, wave estimates for linear wave equations, and spectral multiplier problems associated with elliptic and sub-elliptic linear differential operators are closely related. The aim of this thesis is to further explore these connections and to contribute to a deeper understanding of these phenomena. We focus on certain classes of differential operators which are sub-Laplacians defined as divergence form operators associated with a two-step sub-Riemannian structure on a smooth manifold. The analysis of these operators is closely related to their underlying sub-Riemannian geometry, which is a major challenge in understanding analytic properties of these differential operators. The specific question addressed in this thesis is as follows. The functional calculus for the sub-Laplacian L provided by the spectral theorem allows to define the operator F(L) for every spectral multiplier F : R {\textrightarrow} C. The L^p-spectral multiplier problem asks to identify spectral multipliers F for which F(L) extends to a bounded operator on the Lebesgue space L^p. Usually this question is answered by so-called Mikhlin--H{\"o}rmander type theorems, which require a smoothness condition on the multiplier F. In this thesis we prove spectral multiplier theorems where this smoothness condition is even p-specific. This is done for two specific classes of sub-Laplacians, namely Grushin operators and left-invariant sub-Laplacians on certain subclasses of two-step stratified Lie groups. The proof of these spectral multiplier theorems relies on a careful analysis of the underlying sub-Riemannian geometry and exploiting appropriate restriction type estimates, an idea that goes back to C. Fefferman. A novelty in the restriction type estimates proved in this thesis is an additional truncation along the spectrum of a Laplacian on the second layer of the associated two-step structure.}, url = {https://macau.uni-kiel.de/receive/macau_mods_00004413}, file = {:https://macau.uni-kiel.de/servlets/MCRFileNodeServlet/macau_derivate_00005656/Lars_Niedorf.pdf:PDF}, language = {en} }