A local relative trace formula for PGL(2)

Following a scheme inspired by recent results of B. Feigon, who obtained what she called a local relative trace formula for PGL₂ and a localKuznetsov trace formula for U(2), we describe the spectral side of a local relative trace formula for G :=PGL(2, E) relative to the symmetric subgroup H := PGL(2, F) where E/F is an unramified quadratic extension of local nonarchimedean fields of characteristic 0. The spectral side is given in terms of regularized normalized periods and normalized C-functions of Harish- Chandra. Using the geometric side of the local relative trace formula obtained in a more general setting by the authors and S. Souaifi, we deduce a local relative trace formula for G relative to H. We apply our result to invert some orbital integrals.