Special representations of rank one orthogonal groups

We study the image of the theta correspondence from {Mathematical expression} to a rank one orthogonal group (over a number field). The image consists of cusp forms, the Fourier coefficients of which satisfy a certain invariance property. We show that this property characterizes the image. The proof requires first an analogous local statement (almost everywhere) and then a use of certain Rankin-Selberg integrals.