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

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 ($\mathrm{\partial }$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.