A local relative trace formula for PGL(2)

Following a scheme inspired by B. Feigon, we describe the spectral side of a local relative trace formula for $G:=PGL(2,\mathrm{E})$ relative to the symmetric subgroup $H:=PGL(2,\mathrm{F})$ where $\mathrm{E}\mathrm{/}\mathrm{F}$ is an unramified quadratic extension of local non archimedean fields of characteristic $0$. This spectral side is given in terms of regularized normalized periods and normalized $C$-functions of Harish-Chandra. Using the geometric side obtained in a more general setting by P. Delorme, P. Harinck and S. Souaifi , we deduce a local relative trace formula for $G$ relative to $H$. We apply our result to invert some orbital integrals.