  According to the Langlands functoriality conjecture, broadened to the setting
of spherical varieties (of which reductive groups are special cases), a map
between L-groups of spherical varieties should give rise to a functorial
transfer of their local and automorphic spectra. The "Beyond Endoscopy"
proposal predicts that this transfer will be realized as a comparison between
the (relative) trace formulas of these spaces.
  In this paper we establish the local transfer for the identity map between
L-groups, for spherical affine homogeneous spaces X=H\G whose dual group is
SL(2) or PGL(2) (with G and H split). More precisely, we construct a transfer
operator between orbital integrals for the (X x X)/G-relative trace formula,
and orbital integrals for the Kuznetsov formula of PGL(2) or SL(2). Besides the
L-group, another invariant attached to X is a certain L-value, and the space of
test measures for the Kuznetsov formula is enlarged, to accommodate the given
L-value.
  The fundamental lemma for this transfer operator is proven in a forthcoming
paper of Johnstone and Krishna. The transfer operator is given explicitly in
terms of Fourier convolutions, making it suitable for a global comparison of
trace formulas by the Poisson summation formula, hence for a uniform proof, in
rank one, of the relations between periods of automorphic forms and special
values of L-functions.
