Andrzej Hulanicki

Martin boundary for homogeneous riemannian manifolds of negative curvature at the bottom of the spectrum

In this paper we treat noncoercive operators on simply connected homogeneous manifolds of negative curvature.

Asymptotic Behavior of Poisson Kernels on NA Groups

On a Lie group S = NA, that is a split extension of a nilpotent Lie group N by a one-parameter group of automorphisms A, a probability measure μ is considered and treated as a distribution according to which transformations s ∈ S acting on N = S/A are sampled. Under natural conditions, formulated some over thirty years ago, there is a μ-invariant measure m on N. Properties of m have been intensively studied by a number of authors. The present article deals with the situation when μ(A) = ℙ(s t ∈ A), where ℝ+ ∋ t → s t ∈ S is the diffusion on S generated by a second order subelliptic, hypoelliptic, left-invariant operator on S. In this article the most general operators of this kind are considered. Precise asymptotic for m at infinity and for the Green function of the operator are given. To achieve this goal a pseudodifferential calculus for operators with coefficients of finite smoothness is formulated and applied.

Admissible convergence for the Poisson-Szegö integrals

We prove almost everywhere semirestricted admissible convergence of the Poisson-Szegö integrals ofLp functions (1 <p ≤ ∞) on the Bergman-Shilov boundary of a Siegel domain. In the case of symmetric domains our theorem can be deduced from the results by Peter Sjögren on admissible convergence to the boundary of Poisson integrals on symmetric spaces, although semirestricted admissible convergence means here a more general approach to the boundary then originally defined for symmetric spaces.

MAXIMUM BOUNDARY REGULARITY OF BOUNDED HUA-HARMONIC FUNCTIONS ON TUBE DOMAINS

In this article we prove that bounded Hua-harmonic functions on tube domains that satisfy some boundary regularity condition are necessarily pluriharmonic. In doing so, we show that a similar theorem is true on one-dimensional extensions of the Heisenberg group or equivalently on the Siegel upper half-plane.