University of Birmingham > Talks@bham > Analysis seminar > Spectral multipliers and wave equation for sub-Laplacians

Spectral multipliers and wave equation for sub-Laplacians

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If you have a question about this talk, please contact Diogo Oliveira E Silva.

Mihlin-Hörmander theorem gives the sharp Sobolev order n/2 for a spectral multiplier of the Laplacian to define a bounded operator on L^p for all p∈(1,∞).

We study the same type of statements for sub-Laplacians, which are sub-elliptic operators defined on sub-Riemannian manifolds. It is known that the homogeneous dimension Q can play the same role as n on certain sub-Riemannian manifolds. However, there are examples, such as the Heisenberg groups, where Q>n but n/2 is again the sharp Sobolev order. Here, n is the topological dimension. These results led to conjecture that the sharp Sobolev order for a Mihlin-Hörmander theorem is half the topological dimension in a large class of sub-Riemannian manifolds, e.g., Carnot groups.

We have proven that in no sub-Riemannian manifold the sharp Sobolev order can be lower than half the topological dimension. For the proof, we construct a partial Fourier integral representation of the sub-Riemannian wave propagator.

This is a joint work with Alessio Martini and Detlef Müller.

This talk is part of the Analysis seminar series.

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