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Subelliptic Operators and Harmonic Analysis


School of Mathematics

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Dr A Martini Applications accepted all year round Competition Funded PhD Project (European/UK Students Only)

About the Project

Several problems and results of harmonic analysis are related to the Laplace operator, its spectral theory and functional calculus. In fact, it is often possible to replace the Laplace operator with a more general elliptic operator. However, in various contexts, particularly on noncommutative Lie groups and sub-Riemannian manifolds, the natural substitute for the Laplace operator need not be elliptic, and it may be just subelliptic. Here new and interesting phenomena arise, many basic questions are far from being completely understood, and the known results typically involve a combination of tools from different areas of mathematics, such as evolution equations, representation theory, differential geometry.

References

V. Casarino, M.G. Cowling, A. Martini and A. Sikora, Spectral multipliers for the Kohn Laplacian on forms on the sphere in Cn. Journal of Geometric Analysis 27 (2017), 3302-3338. doi:10.1007/s12220-017-9806-3

A. Martini and D. Müller, Spectral multipliers on 2-step groups: topological versus homogeneous dimension. Geometric and Functional Analysis 26 (2016), 680-702. doi:10.1007/s00039-016-0365-8

A. Martini, F. Ricci and L. Tolomeo, Convolution kernels versus spectral multipliers for sub-Laplacians on groups of polynomial growth, Journal of Functional Analysis 277 (2019), 1603-1638. doi:10.1016/j.jfa.2019.05.024

D. Müller and A. Seeger, Sharp Lp bounds for the wave equation on groups of Heisenberg type, Analysis & PDE 8 (2015), 1051-1100. doi:10.2140/apde.2015.8.1051


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