  We establish explicit Ichino's formulae for the central values of the triple
product $L$-functions with emphasis on the calculations for the real place. The
key ingredient for our computations is Proposition 6.8 which generalizes a
result of Michel-Venkatesh. As an application we prove the optimal upper bound
of a sum of restricted $L^2$-norms of the $L^2$-normalized newforms on certain
quadratic extensions with prime level and bounded spectral parameter following
the methods of Blomer.
