Poisson transform on \(\mathbb{H}^3\) (Q2704088)

Let \(H^3\) be the hyperbolic space realized as the open unit ball in \(R^3\) with the relevant Riemannian structure and the boundary \(S^2\). The Poisson transform is defined by NEWLINE\[NEWLINE (P_\lambda f)(x)=\int_{S^2} \left ( \frac {1-|x|^2}{|x-\omega|^2} \right)^{1+i\lambda/2} f(\omega) d\omega, \quad x \in H^3, f \in L^2(S^2). NEWLINE\]NEWLINE The range of this transform is characterized as the space of eigenfunctions of the Laplace operator on \(H^3\) satisfying certain norm inequalities.The Cotlar-Stein lemma and Y. Meyer's techniques of \(\alpha\)-molecules are used.NEWLINENEWLINEFor the entire collection see [Zbl 0956.00041].