Constructions of representations of solvable Lie groups and non-commutative Fourier analysis

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This research mainly concerns non-commutative Fourier transforms on Lie groups from the following viewpoints. For an exponent p (1＜p≦2) and its conjugate q, the L^p-Fourier transform is defined to be a bounded operator from the space of L^p-functions on the group to the L^q-space of operator fields on the unitary dual; the determination of its norm is one of the main problems of harmonic analysis. We obtained the norm for the groups defined by compact extensions of R^n. We also obtained an estimate of the norm for general connected nilpotent Lie groups. Next, we have treated the Fourier transform as a homomorphism from the C^*-algebra of the group to the C^*-algebra of bounded operator fields over the unitary dual, and began trying to analyze the image of the Fourier transform on a certain example of six dimensional solvable Lie group.