Estimates of automorphic functions

We present a new method to estimate trilinear period for automorphic representations of $SL_2(\mathbb R)$. The method is based on the uniqueness principle in representation theory. We show how to separate the exponentially decaying factor in the triple period from the essential automorphic factor which behaves polynomially. We also describe a general method which gives an estimate for the average of the automorphic factor and thus prove a convexity bound for the triple period.