The Fourier Transform On The Nilpotent Groups Of Step Two And Its Applications In The Complex Analysis

We investigate the Fourier transform on the nilpotent groups of step two, the regular functions and the regular operator on the octonionic Heisenberg group, Kohn’s Laplacian operator. We mainly discuss the regular functions on the oc-tonionic Heisenberg group and find the Szego kernel, the fundamental or relative fundamental solution to the Kohn’s Laplacian on (0, g)-forms on the most non-degenerate CR manifolds of codimension two.In Chapter1, we give a comprehensive survey of the background history and recent developments of group Fourier transform on the nilpotent groups, Kohn’s Laplacian operator and octonionic analysis. And then we introduce some concepts and the main results of the subject.In Chapter2, we mainly give the theory of the irreducible unitary representa-tion, and the theory of the group Fourier transform on the octonionic Heisenberg group, including the Plancherel formula and its extension on the group.In the chapter3, we discuss the octonionic regular functions and the octo-nionic regular operator on the octonionic Heisenberg group. We use the group Fourier transform on the octonionic Heisenberg group and the properties of La-guerre polynomials to analyze the associated Laplacian operator and to construct its kernel. This kernel is exactly the Szego kernel of the orthonormal projection from the space of L2functions to the space of L2regular functions on the octo-nionic Heisenberg group.Chapter4is devoted to the study of Kohn’s Laplacian on (0,q)-forms on the most nondegenerate CR manifolds of codimension two. We obtain the integral representation of the fundamental or relative fundamental solution, and give the proof of the convergence.In the Appendix, we give the explicit forms of the matrices which represent left-multiplying of octonionic algebra and the relation between those matrices and the quaternionic representation matrices.