Transfer formula in the real hyperbolic space \(B^n\) (Q2704090)

Let \(B^n=\{x\in \mathbb{R}^n: |x|<1\}\) be the unit ball and \(S=\{x\in \mathbb{R}^n: |x|= 1\}\) its boundary. We know that the spherical Poisson transform \((P_sT)(x)= \int_sP_s (x,u)dT(u)\) with \(P_s(x,u)= ({1-|x|^2 \over|x-u |^2})^{(n-1) (s+1)/2}\) is a bijection from the space \({\mathcal A}'(s)\) of hyperfunctions on \(S\) onto the spaces \({\mathcal H}_s(B^n)\) of eigenfunctions of the radial Laplace operator of the real hyperbolic ball \(B^n\). On the other hand the Euclidean Poisson transform \(P_eT\) is a bijection from \({\mathcal A}'(s)\) onto the space \({\mathcal H}_e(B^n)\) of Euclidean harmonic functions on \(B^n\). Hence it is natural to look for an explicit expression of the transfer operator \(\theta_s\) such that \(P_s= \theta_s P_e\). In this paper the author uses certain properties of hypergeometric functions to obtain this transfer operator \(\theta_s\): every eigenfunction \(f\) with eigenvalue \((s^2-1)(n-1)^2\) can be written as an integral transform of a unique Euclidean harmonic function \(F\) on \(B^n\): NEWLINE\[NEWLINEf(x)= c(n,s) \int^1_0\left[ {t(1-|x|^2) \over(1-t) (1-t|x|^2)} \right]^{s(n-1)+ 1\over 2} \cdot{F(tx) \over t^{(4-n)/2}} dt.NEWLINE\]NEWLINE As applications of the transfer formula, the author obtains an \(L^p\)-inequality of the spherical Poisson transform and an inequality for spherical functions on the real hyperbolic space.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00041].