Bessel identities in the Waldspurger correspondence over a p-adic field

In this paper we develop the local theory for a Jacquet's relative trace formula. The local theory is essential to the application of the trace formula: it is an identity of Bessel and relative Bessel distributions. We show that the relative Bessel distribution attached to a distinguished (that is, self contragradient) unitary representation of GL₂(k) where k is a p-adic field is given by a locally integrable function, namely the relative Bessel function. We compute the relative Bessel function for principal series, complementary series and special representations. We also show that the Bessel distributions associated to unitary irreducible admissible representations of the double covers of GL₂(k) and SL₂(k) are given by a locally integrable Bessel functions. We compute these Bessel functions for principal series, complementary series and special representations. Finally, we obtain the Waldspurger correspondence via Bessel identities between relative Bessel functions on GL₂(k) and Bessel functions on the double cover of SL₂(k). The Bessel identity also implies the identity between Bessel and relative Bessel distributions.