A characterization of the L2-range of the Poisson transform related to Strichartz conjecture on symmetric spaces of noncompact type

The main purpose of this paper is to solve Strichartz conjecture concerning an image characterization of the Poisson transform in the L2-theory on symmetric spaces of noncompact type. We prove that the Poisson transform provides an isomorphism between the L2-space on the boundary and a certain weighted L2-space consisting of joint eigenfunctions on symmetric spaces of noncompact type. Our approach is based on the scattering theory for the Schrödinger operator. Moreover, we give a Fourier restriction estimate and an asymptotic formula for the Poisson transform.