Bessel identities in the Waldspurger correspondence over the real numbers

We prove certain identities between Bessel functions attached to irreducible unitary representations, of PGL(2)(R) and Bessel functions attached to irreducible unitary representations of the double cover of SL2(R). These identities give a correspondence between such representations which turns out to be the Waldspurger correspondence. In the process we prove several regularity theorems for Bessel distributions which appear in the relative trace formula. In the heart of the proof lies a classical result of Weber and Hardy on a Fourier transform of classical Bessel functions. This paper constitutes the local (real) spectral theory of the relative trace formula for the Waldspurger correspondence for which the global part was developed by Jacquet.