Poisson Transform on ℍ 3

We characterize the image of L ²(S ²) by the Poisson transform P_λ . It is the space of the eigenfunctions of the Laplace operator on the hyperbolic space ℍ 3 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9780429187834/69daaf7f-2564-4f2f-9a9d-3a9774ce28b0/content/eq1722.tif"/> with eigenvalue -(λ ² + 4) satisfying a norm inequality. We use Cotlar–Stein’s lemma and Y. Meyer’s techniques of α-molecules.