Characterisation of the L p Range of the Gen- eralised Poisson Transform of the hyperbolic space B(H n )

TL;DR

This paper characterizes the Lp range of the generalized Poisson transform on hyperbolic spaces over quaternions, linking boundary hyperfunctions to growth conditions of their transforms, with explicit spherical functions.

Contribution

It provides a complete characterization of the Lp range for the generalized Poisson transform on quaternionic hyperbolic spaces, including explicit spherical functions.

Findings

Lp boundary hyperfunctions correspond to transforms satisfying Hardy type growth

Explicit formulas for generalized spherical functions are derived

Characterization applies for p ≥ 2 in quaternionic hyperbolic spaces

Abstract

The aim of this paper is to give the characterisation of the L p Range (p $\ge$ 2) of the Generalised Poisson Transform of the Hyperbolic space B(H n), (n $\ge$ 2), over the classical field of the quaternions H. Namely, if f is an hyperfunction in the boundary of B(H n), then we show that f is in L p ($\partial$B(H n)) if and only if it's generalised poisson transform satisfy an Hardy type growth condition. An explicit expression of the generalized spherical functions is given. Mathematics Subject Classification (2010). Primary 22E46; Secondary 33Cxx.