Admissible and sectorial convergence of generalized Poisson integrals on Harmonic $NA$ groups

TL;DR

This paper establishes a converse Fatou theorem for eigenfunctions of the Laplace-Beltrami operator on Harmonic NA groups, linking sectorial and admissible convergence of Poisson integrals, extending classical results to more general geometric contexts.

Contribution

It extends Fatou type convergence results from classical upper half spaces to Harmonic NA groups and real hyperbolic spaces, providing new insights into eigenfunction boundary behavior.

Findings

Proves a converse Fatou theorem for Harmonic NA groups.

Extends classical convergence results to degenerate hyperbolic spaces.

Establishes sectorial and admissible convergence equivalence for Poisson integrals.

Abstract

We prove a converse of Fatou type result for certain eigenfunctions of the Lalplace-Beltrami operator on Harmonic NA groups relating sectorial convergence and admissible convergence of Poisson type integrals of complex (signed) measures. This result extends several results of this kind proved eariler in the context of the classical upper half space $\mathbb{R}_+^{n+1}$. Similar results are also obtained in the degenerate case of the real hyperbolic spaces.