  Estimates for matrix coefficients of unitary representations of semisimple
Lie groups have been studied for a long time, starting with the seminal work by
Bargmann, by Ehrenpreis and Mautner, and by Kunze and Stein. Two types of
estimates have been established: on the one hand, $L^p$ estimates, which are a
dual formulation of the Kunze--Stein phenomenon, and which hold for all matrix
coefficients, and on the other pointwise estimates related to asymptotic
expansions at infinity, which are more precise but only hold for a restricted
class of matrix coefficients. In this paper we prove a new type of estimate for
the irreducibile unitary representations of $\mathrm{SL}(2,\mathbb{R})$ and for
the so-called metaplectic representation, which we believe has the best
features of, and implies, both forms of estimate described above. As an
application outside representation theory, we prove a new $L^2$ estimate of
dispersive type for the free Schr\"odinger equation in $\mathbb{R}^n$.
