Laplacians on Nonisotropic Heisenberg Groups with Multi-Dimensional Center

We begin with the construction of the nonisotropic Heisenberg group with multi- dimensional center. The sub-Laplacian on it is introduced. By taking the inverse Fourier transform of the sub-Laplacian with respect to the center, we get a family of twisted Laplacians. They are proved to be globally hypoelliptic in the Schwartz space and in the Gelfand-Shilov spaces using the Green functions of the twisted Laplacians. Global regularity in a scale of Sobolev spaces of these twisted Laplacians is given. Equally important are the heat semigroups generated by the twisted Laplacians in terms of Weyl transforms and the Lp Lq estimates of these heat semigroups are presented. Finally, the eigenvalues and eigenfunctions of the twisted bi-Laplacians on these Heisenberg groups are studied in the context of analytic number theory. Global hypoellipticty, essential self-adjointness, Sobolev spaces as well as the Sobolev estimates of the twisted bi-Laplacians are also introduced.