In this paper we study the L^p-convergence of the Riesz means for the sublaplacian on the sphere S^{2n-1} in the complex n-dimensional space C^n. We show that the Riesz means of order delta of a function f converge to f in L^p(S^{2n-1}) when delta>de
lta(p):=(2n-1)|12-1p|. The index delta(p) improves the one found by Alexopoulos and Lohoue, $2n|12-1p|$, and it coincides with the one found by Mauceri and, with different methods, by Mueller in the case of sublaplacian on the Heisenberg group.
We describe recent work on the Bergman kernel of the (non-smooth) worm domain in several complex variables. An asymptotic expansion is obtained for the Bergman kernel. Mapping properties of the Bergman projection are studied. Irregularity properties
of the kernal at the boundary are established. This is an expository paper, and considerable background is provided. Discussion of the smooth worm is also included.
This paper provides a precise asymptotic expansion for the Bergman kernel on the non-smooth worm domains of Christer Kiselman in complex 2-space. Applications are given to the failure of Condition R, to deviant boundary behavior of the kernel, and to L^p mapping properties of the kernel.
