In recent years, effort has been put into following the ideas of M. Ruzhansky and V. Turunen to construct a global pseudo-differential calculus on Lie groups. By this, we mean a collection of operators containing the left-invariant differential calculus with the additional requirement that it be stable under composition and adjunction. Moreover, we would like these operators to have adequate boundedness properties between Sobolev spaces. Our approach consists in using the group Fourier transform to defne a global, operator-valued symbol, yielding pseudo-differential operators via an analogue of the Euclidean Kohn-Nirenberg quantization. The present document treats the case of the Euclidean motion group, which is the smallest subset of Euclidean affne transformations containing translations and rotations. As our representations are infnite-dimensional, the proofs of the calculus properties are more naturally carried out on the kernel side, which means that particular care is required to treat the singularity at the origin. The key argument is a density result which allows us to approximate singular kernels via smooth ones and is proved herein via purely spectral arguments without using classical estimates on the heat kernel.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:702838 |
| Date | January 2016 |
| Creators | Nguyen, Binh-Khoi |
| Contributors | Ruzhansky, Michael |
| Publisher | Imperial College London |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/10044/1/44081 |
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