The Heat Kernels And Their Applications On The Quaternionic And The Cayley Heisenbery Groups

Heisenberg type groups are significant kinds of nilpotent Lie groups of step two and attract attentions of many scholars ([3][4][6][22] etc.) In this thesis we investigate the heat kernels and their applications of the classical Heisenberg type groups which are the quaternionic Heisenberg groups and Cayley Heisenberg groups. Many problems of the geometry and analysis on Lie groups depend on the explicit expressions of the heat kernels for the Lie groups. Therefore, it is considerable to derive explicit expressions of the heat kernels for Lie groups. Hulanicki([21]) and Gaveau([17]) have obtained the explicit expression of the heat kernel on the Heisenberg groups. Then many problems about the Heisenberg groups have been investigated by means of the explicit expression of the heat kernel ([9][16] [20][22][34]etc.). Though Gaveau([17])gave a formula of the heat kernels for free nilpotent Lie groups of step two and Cygan([ll]) obtained a formula of the heat kernels for all nilpotent Lie groups of step two, neither Gaveau's formula nor Cygan's formula are as explicit as that of the Heisenberg groups. Until last year, Professor Zhu Fuliu has derived the explicit expression of the heat kernels for the quaternionic Heisenberg groups([40]).First we apply the explicit expression of the heat kernels given by [40] to get the asymptotic estimations of the heat kernels arid some harmonic functions on the quaternionic Heisenberg groups. Then we use these the estimations to study the Martin compactification of the quaternionic Heisenberg groups. We prove that the Martin boundary of the quaternionic Heisenberg groups is homeomorphic to the unit ball in the quaternionic division algebra H, and the minimal Martin boundary is homeomorphic to the unit sphere S3. Since H is not commutative, structures over H are much more complicate than those over the complex field C. In order to construct the Martin compactification of the Heisenberg groups, one could use the principal circle bundle over S1, while for the quaternionic Heisenberg groups, we must take the unit sphere S3 to replace S1, and each fibre of the principal spherical bundle over S3 is not commutative. As the dimension of the centers of the quaternionic Heisenberg groups is higher than that of the Heisenberg groups, to construct the Martin compactization of the quaternionic Heisenberg groups is considerably difficult.Second we study the Cayley Heisenberg groups. Using Gaveau's stochastic integral method, we obtain the explicit expressions of the heat kernels for the Cayley Heisenberg groups, which closely resembles those of the Heisenberg groups and the quaternionic Heisenberg groups. Since the Cayley division algebra O is neither commutative nor associative, it is much more extremely difficult to derive the explicit expression of the heat kernels for the Cayley Heisenberg groups, which deals with the integrals on the 7-dimensional space, the dimension of the centers of the Cayley Heisenberg groups is higher than ones of Heisenberg groups and the quaternionic Heisenberg groups. As calculate matrices by symbolic calculation of Matlab (a computer software), we verify the correctness of the formula of the heat kernels for the Cayley Heisenberg groups. The verification is independent of computer. As we know, apart from the Heisenberg groups and the quaternionic Heisenberg groups, the Cayley Heisenberg groups are the only kind of nilpotent Lie groups of step two on which an explicit expression of the heat kernelshas been obtained up to now.Finally we consider applications of the heat kernels for the Cayley Heisenberg groups. Being the same as the Heisenberg groups and the quaternionic Heisenberg groups, the explicit expression has many interesting applications, for examples, derive the formula of the Green functions on the Cayley Heisenberg groups, prove the uniform boundedriess of the Riesz transform on the Cayley Heisenberg groups, study the regularity of boundary points in the Dirichlet problem for the heat equation on the Cayley Heisenberg groups, and describe the Mar...