No abstract

No abstract

In this article, we give an explicit calculation of the partial Fourier transform of the fundamental solution to the b -heat equation on quadric submanifolds M ⊂ C n × C m . As a consequence, we can also compute the heat kernel associated with the weighted ∂-equation in C n when the weight is given by exp(−φ(z, z) · λ) where φ : C n × C n → C m is a quadratic, sesquilinear form and λ ∈ R m . Our method involves the representation theory of the Lie group M and the group Fourier transform.

No abstract

In this paper we consider the Hartogs-type extension problem for unbounded domains in C 2 . An easy necessary condition for a domain to be of Hartogs-type is that there is no a closed (in C 2 ) complex variety of codimension one in the domain which is given by a holomorphic function smooth up to the boundary. The question is, how far this necessary condition is from the sufficient one? To show how complicated this question is, we give a class of tube-like domains which contain a complex line in the boundary which are either of Hartogs-type or not, depending on how the complex line is positioned with respect to the domain.