000K  utf8
1100  $c2024
1500  eng
2050  urn:nbn:de:gbv:8:3-2024-00132-3
3000  Niedorf, Lars
4000  Spectral multipliers on two-step structures$hChristian-Albrechts-Universität zu Kiel  [Niedorf, Lars]
4030  Kiel$nChristian-Albrechts-Universität zu Kiel
4209  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 → 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–Hörmander 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.
4950  https://nbn-resolving.org/urn:nbn:de:gbv:8:3-2024-00132-3$xR$3Volltext$534
4961  https://macau.uni-kiel.de/receive/macau_mods_00004413
5051  510
5550  Grushin operator
5550  Lie group
5550  restriction type estimate
5550  spectral multiplier
5550  sub-Laplacian
5550  sub-Riemannian geometry
5550  two-step structure